Chapter II
The Fundamental Ideas of the Integral and Differential Calculus
Among the limiting processes of analysis, there are two processes with an especially important role, not only because they arise in many different connections, but chiefly due to their very close reciprocal relationship. Isolated examples of these two limiting processes, differentiation and integration, have even been considered in classical times; but it is the recognition of their complementary nature and the resulting development of a new and methodical mathematical procedure which marks the beginning of the real systematic differential and integral calculus. The credit of initiating this development belongs equally to the two great geniuses of the Seventeenth Century, Newton and Leibnitz, who, as we know to-day, made their discoveries independently of each other. While Newton, in his investigations, may have succeeded in stating his concepts more clearly, Leibnitz's notation and methods of calculation were more highly developed; even to-day, these formal portions of Leibnitz's work form an indispensable element in the theory.
We first encounter the integral in the problem of measuring the area of a plane region bounded by curved lines. Then, more refined considerations permit us to separate the notion of integral from the naïve intuitive idea of area and to express it analytically in terms of the notion of number only. We shall find this analytical definition of the integral to be of great significance not only because it alone enables us to attain complete clarity in our concepts, but also because its applications extend far beyond the calculation of areas. We shall begin by considering the matter intuitively.
2.1.1 The integral as an Area: Let there be are given a function f(x), which is continuous and positive in an interval, and two values a and b (a < b) in that interval. We think of the function as being represented by a curve and consider the area of the region which is bounded above by the curve, at the sides by the straight lines x = a and x = b and below by the portion of the x-axis between the points a and b (Fig. 1). That there is a definite meaning to speaking of the area of this region is an assumption inspired by intuition, which we state here
expressly as a hypothesis. We call
this area Fab
the definite
integral of the function f(x) between
the limits a and b. When
we actually seek to assign a numerical value to this area, we
find that we are, in general, unable to measure areas with curved
boundaries, but we can measure polygons with straight sides by
dividing them into rectangles and triangles. Such a sub-division
of our area is usually impossible. It is, however, only a short
step to conceive in the following manner the area as the limiting
value of a sum of areas of rectangles. We subdivide the part of
the x-axis between a and b into n
equal parts and erect at each point of sub-division the ordinate
up to the curve; the area is thus divided into n strips.
We can no more calculate the area of such strips than we could
that of the original surface; but if, as shown in Fig.
2, we find first the least and then the greatest value of the
function f(x) in each sub-interval and then
replace the corresponding strip (1) by a rectangle with height
equal to the least value of the function, and (2) by a rectangle
with height equal to the greatest value of the function, we
obtain two step-shaped figures. (In Fig. 2 above, the
first of these is drawn with a solid line, the second with a
broken line.) The first step-shaped figure obviously has an area
which is at most equal to the area Fab
which we are trying to determine; the second has an area which is
at least as large as Fab.
If we denote the sum of the areas of the first set of rectangles
by 
(lower sum) and the sum of the areas of the second set by 
(upper sum), we find
If we now make the subdivision
finer and finer, i.e., let n increase without limit,
intuition tells us that the quantities and approach closer and
closer to each other and tend to the same limit Fab.
We may therefore consider our integral as the limiting value
Intuition also tells us the possibility of an immediate generalization. It is by no means necessary that the n sub-intervals should all be of the same length. On the contrary, they may have different lengths provided only that, as n increases, the length of the longest sub-interval tends to 0.
2.1.2 The Analytical Definition of the Integral: In the above section, we have considered the definite integral as a number given by an area, hence to a certain extent as previously known, and have subsequently represented it as a limiting value. We shall now reverse this procedure. We no longer take the point of view that we know by intuition how an area can be assigned to the region under a continuous curve or, indeed, that this is possible; we shall, on the contrary, begin with sums formed in a purely analytical way, like the upper and lower sums defined previously, and shall then prove that these sums tend to a definite limit. We take this limiting value as the definition of the integral and of the area. We are naturally led to adopt the formal symbols which have been used in the integral calculus since Leibnitz's time.
Let f(x) be a function which is positive and continuous in the interval a £ x £ b (of length b - a). We think of the interval as being sub-divided by (n—1) points x1, x2, ··· , xn-1 into n equal or unequal sub-intervals and, in addition, we let x0=a, xn=b. In each interval, we choose a perfectly arbitrary point, which may be within the interval or at either end; suppose that in the first interval we choose the point x1, in the second one the point x2, ··· and in the n-th interval the point xn. Instead of the continuous function f(x), we now consider a discontinuous function (step-function) which has the constant value f(x1) in the first sub-interval, the constant value f(x2) in the second sub-interval, ··· , the constant value f(xn) in the n-th sub-interval. As is shown in Fig. 3, the graph of
this step-function defines a series of rectangles, the sum of the areas of which is given by
This expression is usually abbreviated by means of the summation sign S:
by introducing the symbol
we can simplify this formula to
(Here the symbol D is not a factor, but denotes the word difference. By definition, the entire inseparable symbol Dxn means the length of the n-th sub-interval.) Our basic assertion may now be stated as follows:
If we let the number of points of sub-division increase without limit and at the same time let the length of the longest sub-interval tend to 0, then the above sum tends to a limit. This limit is independent of the particular manner in which the points of division x1, x2, ··· , xn-1 and the intermediate point x1, x2, ··· , xn-1 are. chosen.
We shall call this limiting value the definite integral of the function f(x), the integrand, between the limits a and b; as we have already mentioned, we shall consider it as the definition* of the area under the curve y = f(x) for a £ x £ b. Our basic assertion may then be re-worded: If f(x) is continuous in a £ x £ b, its definite integral between the limits a and b exists.
* Of course, we may also define the notion of are in a purely geometrical way and then prove that such a definition is equivalent to the above limit=definition (cf. 5.2.1).
This theorem on the existence of the definite integral of a continuous function can be proved by purely analytical methods and without appeal to intuition. We shall nevertheless pass it over for the present and return to it in A2.1 after the use of the concept of integral has stimulated the reader's interest in constructing for it a firm foundation. For the moment, we shall content ourselves with the fact that the intuitive considerations above have made the theorem appear to be extremely plausible.
2.1.3 Extensions, Notation. Fundamental Rules: The above definition of the integral as the limit of a sum led Leibnitz to express the integral by the symbol
The integral symbol is a modification of a summation sign which has the shape of a long S. The passage to the limit from a sub-division of the interval into finite portions Dxn is suggested by the use of the letter d in place of D. However, we must guard ourselves against thinking of dx as an infinitely small quantity or infinitesimal, or of the integral the sum of an infinite number of infinitely small quantities. Such a concept would be devoid of any clear meaning; it is only a naïve interpretation of what we have previously carried out with precision.
In the above figures, we have assumed: (1) that
the function f(x) is positive throughout the
interval, and (2) that b > a. However,
the formula which defines the integral as the limit of a sum is
independent of any such assumptions. For if f(x) is
negative in all or a part of our interval, the only effect is to
make the corresponding factors f(xn) in
our sum negative instead of positive. We shall naturally assign
to the region bounded by the part of the curve below the x-axis
a negative area, which is in agreement with the familiar
convention of sign in 
analytical
geometry. The total area bounded by a curve will thus, in
general, be the sum of positive and negative terms, corresponding
to the portions of the curve above and below the x-axis,
receptively.*
* For the area of regions bounded by arbitrary closed curves, cf. 5.2.2 .
If we also omit the condition a < b and assume that a > b, we can still retain our arithmetic definition of integral; the only change is that, if we traverse the interval from a to b, the differences Dxn are negative. We are thus led to the relation
which holds for all values of a and b
(a ¹ b). Correspondingly, we define 
as equal
to zero.
Our definition immediately yields the basic relation (cf. Fig. 4 above):
for a < b < c. By means of the preceding relations, we find at once that this equation is also true for any relative positions of the end-points a, b, x .
We obtain a simple but important fundamental rule by considering the function cf(x), where c is a constant. From the definition of the integral, we immediately obtain
Moreover, we assert the addition rule: If
then
The proof is quite simple.
We add a final remark about the variable
of integration, which is
perfectly obvious, but very important in applications. We have
written our integral in the form . During evaluation of the integral, it does not matter
whether we use the letter x or any other letter to
denote the abscissae of the co-ordinate system, i.e., the independent
variable. We use this
particular symbol for the variable of integration and it is
therefore a matter of complete indifference; instead of 
, we could
equally well write 
or any similar expression.
We are now in a position to carry out the limiting process prescribed by our definition of the integral and thus actually calculate the area in question in a number of special cases; we shall do this in a series of examples, where (except in No. 5) we shall employ only the upper or the lower sum.*
*We have it as a useful exercise for the reader to prove that in the following examples we actually do arrive at the same result whether we use the upper or the lower sum.
2.2.1. Integration of a Linear Function:
We first consider the function f(x) = xn, where n is an integer greater than or equal to 0. For n = 0, i.e. for f(x) = 1, the result is so obvious that we simply write it down:
For the function f(x) = x, the integration is again a triviality from the geometrical point of view. Its integral
is simply the area of the trapezoid shown in Fig. 6 below, which by an elementary formula is
We shall now verify that our limiting process leads to exactly the same result. In calculating the limit, we can restrict ourselves to the discussion of upper sums or lower sums. We subdivide the interval from a to b into n equal parts by means of the points of sub-division
where h = (b - a)/n. The integral must then be the limit of the following sum, which is an upper sum, if b < a and a lower sum if b > a:
By an elementary formula, we have
and our expression may therefore be rewritten in the form
As n increases, the right hand side obviously tends to the limit
as was to be proved.
A problem which is not quite as simple as that of the integration of the function f(x) = x², or, in geometrical language, of determining the area of the region bounded by a segment of a parabola, a segment of the x-axis and two ordinates. For example, consider the integral
where b ³ 0 (Fig. 6) and sub-divide the interval 0 £ x £ b into n equal parts of length h = b/n; the area which we then wish to find is the limit of the expression (upper sum):
However, the sum in brackets has already been found (cf. 1.4, footnote)!
If we substitute this expression and rewrite the result in a slightly different form, our sum becomes
As n increases beyond all bounds, this expression tends to the limit b³/3 and we obtain the required integral
Using the general result above, we immediately derive the formula
2.2.3 Integration of xa, where a is any positive integer: As a third example, we consider the integration of the function
where a is any positive integer. For the computation of the integral
,
(where we assume 0 < a < b),
it would be inconvenient to subdivide the interval into n
equal parts.* However, the passage to the limit may be
accomplished very easily, if we subdivide in geometric
progression in the following
manner. Let 
and subdivide the interval by the points
* We should then be obliged to base the evaluation of the integral upon the calculation of the limit of
as n ® ¥; the reader may work this result out as has been indicated in the footnote referred to above.
The required integral is then the limit of the sum:
The terms in the last bracket form a geometric progression with the common ratio qa+1 ¹ 1. If we sum this progression, we obtain for the entire expression the value
We now replace q by its value (b/a)1/n; our sum then takes the form
If we now let n increase without limit, the first factor retains its value. Since q¹1, we can use the formula for the sum of a geometric progression and write the second factor in the form
as the equation q = (b/a)1/n shows that q tends to 1 as n ® ¥, the second factor will have the limit l/(a + 1). Thus, finally, the value of our integral is given by
In principle,the above calculation is simple, but itsdetails are somewhat complicated. We shall later on discover that it can be entirely avoided once we have become better acquainted with integration theory.
2.2.4 Integration of xa, where a is any Rational Number other than -1:
The result obtained above may be generalized considerably without essential complication of the method. Let a = r/s be a positive rational number, r and s positive integers; in the evaluation of the integral given above, nothing is changed except the evaluation of the limit (q - 1)/(qn+1 - 1) as q approaches 1. This expression is now simply (q - 1)/(q(r+s)/s - 1). Let us set q1/s = t (t ¹ 1); then, as q tends to 1, t will also tend to 1. We have therefore to find the limiting value of (ts - 1)/(tr+s - 1) as t approaches 1. If we divide both the numerator and denominator by t - 1 and transform them as before, the limit simply becomes
Since both the numerator and denominator are continuous in t, this limit is at once determined it we set t = 1. We thus arrive at the limit s/(r + s) = 1/(a + 1) and obtain for every positive rational value of a
This formula remains valid for negative rational values of a, provided we exclude the value a = -1 for which the formula used above for the sum of the geometric progression loses its meaning. We must now investigate the limit of the expression (q - 1)/(qn+1 - 1) for negative values of a, say a = -r/s. For this purpose, we set q-1/s = t and obtain
Hence we seek to determine the limiting value of
We leave it to the reader to prove that this limit is again equal to 1/(a + 1), that is, that we have the integral
for the general case of rational values of either positive or negative a, with the exception of a = - 1.
The form of the right-hand side of this equation shows that the expression is not valid for a = -1, since both the numerator and denominator would then be zero.
It is natural to assume that the range of validity of our last formula extends also to irrational values of a. We shall actually establish this in 2.7.2 by a simple passage to the limit.
2.2.5 Integration of sin x and cos x: As a last example, consider the function f(x) = sin x. Also in this case, we shall employ a special device. We express the integral
as the limit of the sum:
where h = (b - a)/n. We multiply the right-hand bracket by 2 sin h/2 and recall the well-known trigonometrical formula
provided h is not a multiple of 2p, we obtain
Since a + nh = b, the integral becomes the limit of
Now we know from Chapter I that as h tends to 0, the expression h/2/sin h/2 approaches the limit 1. The desired limit is then simply cos a - cos b and we thus arrive at the integral
In the same manner, the reader may verify the formula
Almost every one of these examples has been attacked by means of some special method or particular device. The essential point of the systematic Integral and Differential Calculus is the very fact that we utilize instead of such special devices considerations of a general character which lead us directly to the desired result. In order to arrive at these methods, we must now turn our attention to the other fundamental concept of higher analysis, the derivative.
1. Find the area bounded by the parabola y = 2x³ + x + 1, the ordinates x = 1 and x = 3 and the x-axis.
2. Find the area bounded by the parabola y = ½x² + x + 1I and the straight line y = 3 + x.
3. Find the area bounded by the parabola y² = 5x and the straight line y = 1+ x.
4. Find the area bounded by the parabola y = x² and the straight line y = ax + b.
5. Using the methods developed above, evaluate the integrals
where a is an arbitrary integer.
6. Use the formulae obtained in Example 5 along with the identities
to prove that
7. Use Exercise 1. in 1.4 to evaluate
by division into equal sub-intervals.
8. Evaluate
(where n is an integer) by expansion of the bracket.
The concept of the derivative, like that of the integral, has an intuitive origin. Its sources are: (1) the problem of construction of the tangent to a given curve at a given point and (2) the problem of finding a precise definition for the velocity during an arbitrary motion.
2.3.1 The Derivative and the Tangent: We shall first deal with the tangent problem. If P is a point on a given curve (Fig. 7), we shall, in conformity with naive intuition, define the tangent to the curve at the point P by means of the following geometrical limiting process. In addition to the point P, we consider a second point P1 on the curve. Through the two points P and P1, we draw a straight line, a secant of the curve. If we now let the point P1 move along the curve towards the point P, this secant will tend to a limiting position which is independent of the direction from which it approaches P. This limiting position of the secant is the tangent and the statement that such a limiting position of the secant exists is equivalent to the assumption that the curve has a definite tangent or a definite direction at the point P. (We have used here the word assumption because we have actually made one. The hypothesis that the tangent exists is valid for most simple curves, but is by no means true for all curves or even for all continuous curves.)
Once we
have represented our curve by means of a function y = f(x),
there arises the problem of representing our geometrical limiting
process analytically, using the function f(x).
We take the angle, which a straight line l makes with
the x-axis, as being the angle through which the
positive x-axis must be turned in the positive direction* in
order to be for the first time parallel to the line l.
Let a1 be the angle which the secant PP1
forms with the positive x-axis (Fig, 7) and a the angle
which the tangent forms with the positive x-axis. Then,
disregarding the case of a perpendicular tangent, we obviously
have
where the meaning of the symbols is perfectly clear. If x, y (= f(x)) and x1, y1 (=f(x1)) are the coordinates of the points P and P1, respectively, we immediately find**
and thus our limiting process is represented by the equation
* That is, in such a direction that a rotation of p /2 brings it into coincidence with the positive y-axis, in other words, counter-clockwise.
** In order that this equation may have a meaning, we must assume that 0 <|x-x1| < d, d being sufficiently small. In what follows, corresponding assumptions will often be made tacitly in the steps leading to limiting processes.
The expression
is called the difference quotient of the function y =f(x), since the symbols Dy and Dx denote the differences of the function y =f(x) and of the independent variable x, respectively. (As in 2.1.2, the symbol D is an abbreviation for the difference and is not a factor!) The tangent of a, the direction angle of the curve,* is therefore equal to the limit to which the difference quotient of our function tends when x1 tends to x.
* The slope or gradient of the curve is given by tan a, whence also the term gradient is used for the derivative of the function represented by the curve.
We call this limit the derivative or differential coefficient of the function y=f(x) at the point x and, as Lagrange did, denote it by the symbol y' =f '(x) or, as Leibnitz did, the symbol dy/dx or df(x)/dx or d/dx f(x).* On p. 100, we shall discuss the meaning of Leibnitz's notation in greater detail; here we just point out that the notation f '(x) expresses the fact that the derivative is itself a function of x, since it has a definite value for each value of x in the interval under consideration. This fact is sometimes emphasized by the use of the terms derived function, derived curve (cf. 2.3.6). We again quote the definition of the derivative:
or
where we have replaced in the last expression x1 by x + h.
*Cauchy's notation Df(x) if also occasionally found in the literature.
It is impossible to find the derivative merely by putting x1 = x in the expression for the difference quotient, for then the numerator and denominator would both be equal to 0 and we should then led to the meaningless expression 0/0. On the contrary, the actual performance of the passage to the limit in each individual case depends on certain preliminary steps (transformation of the difference quotient).
For example, for the function f(x) = x², we have
This function x1 + x
is not the same function as 
, because the function x1
+ x is defined at one point where the quotient is not
defined, namely, at the point x1 = x.
For all other values of x1, the two functions
are equal to each other, whence in the above passage to the
limit, in which we specifically required that x1
¹ x,
we obtain the same value 
However, since the function x1
+ x is defined and continuous at the point x1
= x, we can do with it what we could not do with the
quotient, namely, pass to the limit by simply putting x1
= x. We then obtain for the derivative the expression
The performance of such a process, i.e., the actual formation of the derivative is called the differentiation of the function f(x). We shall see later on how this process of differentiation can actually be carried out in all important cases.
Now, the fact that the problem of differentiating a given function has a definite meaning apart from the geometrical intuition of the tangent is of great significance. The reader will recall that, in the case of the integral, we freed ourselves from the geometrical intuition of area and, on the contrary, based the notion of area on the definition of the integral. Now, independently of the geometrical representation of a function y = f(x) by means of a curve, we shall define the derivative of the function y = f(x) as being the new function y' = f'(x), given by the equation above, provided always that the limit of the difference quotient exists. If this limit exists, we say that the function f(x) is differentiable. From now on, we shall assume that every function dealt under consideration is differentiable unless specific mention is made to the contrary.* It should be observed that, if the function f(x) is to be differentiable at the point x, the limit as h ® 0 of the quotient [f(x + h) - f(x)]/h must exist independently of the manner in which h tends to 0, whether through positive or through negative values or without restriction as to the sign.
* Examples of cases in which this assumption is not satisfied will be given later (2.3.5).
Once we have found the derivative f '(x), we take the direction which makes an angle a with the positive x-axis given by the equation tan a = f '(x) as the direction of the tangent to the curve at the point (x,y). We thus avoid the difficulties which arise out of the indefiniteness of the geometrical view, since we base the geometrical definition on the analytical one and not vice versa.
Nevertheless, the visualization of the derivative as the tangent to the curve is an important aid to understanding, even in purely analytical discussions. Accordingly, we shall at once accept the statement based on geometrical intuition: If f '(x) is positive and the curve is traversed in the direction of increasing x, then the tangent slants upwards, and therefore at the point in question the curve rises as x increases; on the other hand, if f '(x) is negative, the tangent slants downwards and the curve falls as x increases (Fig. 8). Analytically, this follows from the remark that the limit of [f(x + h) - f(x)]/h cannot be positive unless the function is increasing at the point x, by which we mean that for all values of h sufficiently close to 0 the value of f(x + h) is greater or smaller than f(x) according to whether h is positive or negative. We can, of course, make a corresponding statement for the case when f '(x) is negative.
2.3.2 The Derivative as a Velocity: Just as naive intuition led us to the notion of the direction of the tangent to a curve, so it causes us to assign a velocity to a motion. The definition of velocity leads us once again to exactly the same limiting process which we have already called differentiation.
For example, consider the motion of a point along a straight line, the position of the point being determined by a single co-ordinate y, which is the distance, with its proper sign, of our moving point from a fixed point on the line. The motion is given, if we know y as a function of the time t, y = f(t). If this function is a linear function f(t) = ct + b, we speak of uniform motion with the velocity c, and for every pair of values t and t1, which are not equal to each other, we can write
The velocity is therefore the difference quotient of the function ct + b, and this difference quotient is completely independent of the particular pair of instants considered. But what are we to understand by the velocity of motion at an instant t if the motion is no longer uniform?
In order to arrive at this definition, we
consider the difference quotient 
which we shall call the
average
velocity
in the time interval between t1
and t. If now this average velocity tends to a definite
limit when we let the instant t1 come closer
and closer to t, we shall naturally define this limit as
the velocity at the time t. In other words: The
velocity at the time t is the derivative
From this new meaning of the derivative, which in itself has nothing to do with the tangent problem, we see that it really is appropriate to define the limiting process of differentiation as a purely analytical operation independently of geometrical intuitions. Here again, the differentiability of the position-function is an assumption which we shall always make tacitly and which, in fact, is absolutely necessary if the notion of velocity is to have any meaning.
As a simple example of the connection between motion and velocity, consider the case of a freely falling body. We begin with the experimentally established law that the distance traversed in time t by a freely falling body is proportional to t² and therefore can be represented by a function of the form
As before, we find immediately that the velocity is given by the expression f'(t) = 2at, which shows that the velocity of a freely falling body increases proportionally to the time.
2.3.3
Examples: We now proceed to work out a number of examples of the
actual differentiation of functions and begin with the function y
=f(x), where c is a constant. It is
then always true that f(x + h) - f(x)
= c - c = 0, so that 
that is, the derivative of a constant is zero.
For a linear function y = f(x) = cx + b, we find that
Moreover, we shall differentiate the function
at first assuming that a is a positive integer. Provided x1 ¹ x, we have
the right-hand side of this equation is equal to x1a-1 + x1a-2x + ··· + xa-1, as we see either by direct division or by using the formula for the sum of a geometric progression. The new expression for the right-hand side of the equation is a continuous function, whence we can carry out the passage to the limit (x1 ® x) by simply replacing x1 everywhere by x. Each term is then xa-1 and since the number of terms is exactly a, we obtain
We arrive at the same result if a is a negative integer - b ; however, we must assume that x is not zero. We then find that
Once again, we can carry out the passage to the limit simply by replacing x1 ] everywhere by x. Then, just as above, we obtain for the limit the expression
Hence, for negative integral values of a, the derivative is again given by
Finally, we shall prove the same formula, where x is positive and a any rational number. We let a = p/q, where both p and q are integers and, moreover, positive. (If one of them were negative, no essential changes in the proof would be required; for a = 0, the result is already known, since xa is then constant.) We now have
If we now let x1/q = x and x11/q = x1, we obtain
After this last transformation, we can immediately perform the passage to the limit (x1 ® x or, what amounts to the same thing. x1 ® x), and thus obtain for the limiting value
which is formally the same result as before. We leave it to the reader to prove that the same differentiation formula holds also for negative rational superscripts. Once we have developed the theory in a more connected form, we shall return in 2.7.2 to the differentiation of powers.
Finally, as a further example, we consider the differentiation of the trigonometric functions: sin x and cos x. We use the elementary trigonometrical formula
Now, by 1.7, we know that
Thus, we find for the required derivative immediately
The function y = cos x can be differentiated in exactly the same manner. Starting with
and taking the limit as h ® 0, we at once obtain the derivative
2.3.4. Some Fundamental Rules of Differentiation: Just as in the case of the integral, certain simple, but fundamental rules for forming the derivative follow immediately from the definition. If f (x) = f(x) + g(x), then f '(x) = f '(x) + g'(x); again, if y(x) = cf(x) (where c is a constant), then y '(x) = cf '(x). Because
and
our statements follow directly by passage to the limit.
For example, according to these rules, the derivative of the function f(x)=f(x)+ax+b (where a and b are constants) is given by the equation
f '(x)=f '(x)+a.
2.3.5 Differentiability and Continuity of Functions: It is useful to point out that, if we know that a function can be differentiated, we need not give any special proof of its continuity.
If a function is differentiable, then it is necessarily continuous.
In fact, if the difference quotient 
approaches a definite limit as h tends to zero, the
numerator of the fraction, that is, f(x + h)
- f(x), must tend to zero with h, and
this fact expresses the continuity of the function f(x)
at the point x. 
However, the converse of this is definitely false; it is not true that every continuous function has a derivative at every point. The simplest example, which disproves this assumption, is the function f(x) = |x|, i.e., f(x) = - x for x £ 0 and f(x) = x for x ³ 0; its graph is shown in Fig. 9. At the point x = 0, this function is continuous, but has no derivative. The limit of [f(x+h)-f(x)]/h is equal to 1 if h tends to 0 through positive values and equal to - 1 if h tends to zero through negative values; if we do not restrict the sign of h, There does not exist a limit. We say that the function has different right hand and and left-hand derivatives at the point x, where by we mean the limiting values of [f(x+h)-f(x)]/h as h approaches 0 through positive values only and negative values only, respectively. Thus, the differentiability of a function requires not merely that the right-hand and left-hand derivatives exist, but that they are equal. Geometrically, the inequality of the two derivatives means that the curve has a sharp corner.
As further examples of points
where a continuous function is not differentiable, we consider
the points where the derivative becomes infinite, i.e., the
points at which there exists neither a right-hand nor a left-hand
derivative, the difference quotient [f(x+h)-f(x)]/h
increasing beyond all bounds as h ® 0. For example, the
function 
is defined and continuous for all values of x.
For all non-zero values of x, its derivative is given by the formula y' = x-2/3/3.
At the point x = 0, we have [f(x+h)-f(x)]/h
= h1/3/h = h-2/3,
and we see at once that, as h ® 0, the expression has no
limiting value, but tends to ¥. 
This state of affairs is often briefly described by saying that the function possesses an infinite derivative or the derivative ¥ at the point in question; however, we should remember that this merely means that, as h tends to 0, the difference quotient increases beyond all bounds and that the derivative in the sense, in which we have defined it, really does not exist. The geometrical meaning of an infinite derivative is that the tangent to the curve is vertical (Fig. 10 above).
The function 
which is defined and
continuous for x ³
0 , is also non-differentiable at the point x = 0. Since y is
undefined for negative values of x, we here consider
only the right-hand derivative. The equation 
shows us that this
derivative is infinite; the curve touches the y-axis at the
origin (Fig. 11 above).
Finally, the function 
is a case in which the right-hand 
derivative at the point x = 0 is positive and
infinite, while the left-hand derivative is negative and
infinite, as follows from the relation
As a matter of fact, the continuous curve y = x2/3, the so-called semi-cubical parabola or Neil's parabola, has at the origin a cusp, perpendicular to x-axis(Fig. 12).
2.3.6. Higher Derivatives and
their Significance: The derivative f '(x) of a function
is itself a function of x, the graph of which we call
the derived
curve of the given curve. For example, the derived curve of
the parabola y = x² is a straight line,
represented by the function y = 2x. The derivede curve of y =sin
x is y = cos x; similarly, the derived curve of
y = cos x is y = -sin x. (Any
of these latter curves can be obtained from the others by
translation in the direction of the x-axis.
It is now quite a natural step to form the derived curves of the derived curves, i.e., to form the derivative of the function f '(x) = f (x). This derivative
provided that it really exists, we shall call the second derivative of the function f(x) and denote it by f"(x).
Similarly, we may attempt to form the derivative of f "(x), the so-called third derivative of f(x), which we then denote by f "'(x). In the case of most functions of importance, there is nothing to stop us from imagining this process repeated as many times as we like and from thus defining the n-th derivative f (n)(x). At times, it will be convenient to call the function f(x) its own zero-th derivative.
The terms second, third, ··· , n-th differential coefficient are also employed.
If the independent variable is interpreted as the time t and the motion of a point is represented by means of the function f(t), the physical meaning of the second derivative is found to be the velocity with which the velocity f '(t) changes, or, as it is usually called, the acceleration. Later on, we shall discuss in detail the geometrical interpretation of the second derivative . However, we may note here the following facts: At a point where f ''(x) is positive, f '(x) increases with x; on the other hand, if f "(x) is negative, f '(x) decreases with x.
2.3.7 The Derivative and the Difference Quotient: The fact that in the limiting process, which defines the derivative, the difference Dx tends to 0 is sometimes expressed by saying that the quantity Dx becomes infinitely small. This expression indicates that the passage to the limit is regarded as a process during which the quantity Dx is never zero, yet approaches zero as closely as we please. In Leibnitz's notation, the passage to the limit in the process of differentiation is symbolically expressed by replacing the symbol D by the symbol d, so that we can define Leibnitz's symbol for the derivative by the equation
However, if we wish to obtain a clear grasp of the meaning of the differential calculus, we must beware of regarding the derivative as the quotient of two quantities which are actually infinitely small. The difference quotient Dy/Dx definitely must be formed with differences Dx which are not equal to 0. After forming this difference quotient, we must imagine the passage to the limit carried out by means of a transformation or some other device. We have no right to assume that first Dx goes through something like a limiting process and reaches a value which is infinitesimally small but still not 0, so that Dx and Dy are replaced by the infinitely small quantities or infinitesimals dx and dy, and the quotient of these quantities is then formed. Such a conception of the derivative is incompatible with the clarity of ideas demanded in mathematics; in fact, it is altogether meaningless. For a great many simple-minded people, it undoubtedly has a certain charm - the charm of mystery - which is always associated with the word infinite; and in the early days of the differential calculus, even Leibnitz himself was capable of combining these vague mystical ideas with a thoroughly clear understanding of the limiting process. It is true that this fog, which surrounded the foundations of the new science, did not prevent Leibnitz or his great successors from finding the right path. But this does not release us from the duty of avoiding every such hazy idea in our building-up of the differential and integral calculus.
However, the notation of Leibnitz is not merely attractive in itself, but is actually of great flexibility and the greatest use. The reason is that we can deal in many calculations and formal transformations with the symbols dy and dx in exactly the same way as if they were ordinary numbers. They enable us to give neater expression to many calculations which can be carried out without their use. In the following pages, we shall see this fact verified over and over again and shall find ourselves justified in making free and repeated use of it, provided we do not lose sight of the symbolical character of the signs dy and dx.
Leibnitz has also devised for the second and higher derivatives a notation of great suggestiveness and practical utility. He thinks of the second derivative as the limit of the second difference quotient in the following manner. In addition to the variable x, we consider x1 = x + h and x2 = x + 2h. We then take the second difference quotient as meaning the first difference quotient of the first difference quotient, i.e., the expression
where y = f(x), y1 = f(x1) and y2= f(x2). If we also write h = Dx and y2-y1=Dy1, y1-y=Dy, we may appropriately call the expression in the last bracket the difference of the difference of y or the second difference of y and write symbolically *
* Here DD = D² is not a square, but merely a symbol for difference of difference or second difference.
In this symbolic notation, the second difference quotient is then D²y/(Dx)², where the denominator is really the square of Dx, while in the numerator the number 2 syrnbolically denotes the repetition of the difference process. This symbolism for the difference quotient led Leibnitz to introduce the notation
for the second and higher derivatives, and we shall find that this notation also stands the test of use.
We must emphasize that the statement that the second derivative may be represented as the limit of the second difference quotient requires proof. For we previously defined the second derivative not in this way, but as the limit of the first difference quotient of the first derivative. In actual fact, the two definitions are equivalent, provided the second derivative is continuous; however, the proof is not given, as we have no particular need for it here.
2.3.8 The Mean Value Theorem: There exists between the derivative dy/dx=f'(x) and the difference quotient a simple relation which is important for many purposes is known as the mean value theorem and is obtained in the following manner. We consider the difference quotient
of a function f(x)
and assume that the derivative exists everywhere in the interval x1
£ x
£ x2
so that the graph of the curve has everywhere a tangent. The
difference quotient will be represented by the direction of the secant (Fig. 14); in fact, it is the tangent of the angle a shown in
the figure. Let us imagine that this secant is shifted parallel
to itself. At least once, it will reach a position in which it is
a tangent to the curve at a point between x1
and x2, namely at the point of the curve
which is at the greatest distance from the secant. Hence there
will be an intermediate value x such that
This statement is called the mean value theorem of the differential calculus. We can also express it somewhat differently by noticing that the number x may be written in the form
where q is a certain number between 0 and 1. In applications of the mean value theorem, we shall often find that q cannot be determined more accurately than this, but it will usually turn out that a more accurate value is not required. When accurately formulated, the mean value theorem is:
If f(x) is continuous in the closed interval x1 £ x £ x2 and differentiable at every point of the open interval x1 < s < x2, then there is at least one value q, where 0<q <1 such that
If we replace x1 by x and x2 by x + h, we can express the mean value theorem by the formula
We wish to emphasize that, while it is essential that f(x) should be continuous for all points of the interval, including the end-points, we need not assume that the derivative exists at the end-points. This apparently trivial remark is actually useful in many applications.
If at any point in the interior of an interval the derivative fails to exist, the mean value theorem is not necessarily true. This is shown by the example f(x)=|x|.
We can complete
our intuitive argument by the following consideration. There is
at least one point P on the curve which has the greatest
possible distance from the chord joining the points on the curve
with the abcissae x1 and x2
(Fig. 15). At this point, the curve has by assumption a definite
tangent. We shall now prove that this tangent must be parallel to
the chord. By definition, the tangent is the limiting position of
the secant and is obtained by joining P to a point Q
on the curve and letting the point Q move towards P.
Since, by assumption, Q is not further from the chord than
P, the line PQ produced in the direction P
to Q must either intersect the chord or run parallel to
it; and this must be the case, no matter on which side of P
lies the point Q. This, however, is only possible if the
limiting position is parallel to the chord. If we denote the
abscissa of the point P by x, the slope f
'(x) of the tangent at P is then equal to the
slope of the chord, [f(x1) - f(x2)]/[x1
- x2)], whence we may simply take for the
number x in the theorem the abscissa of P.
The rigorous proof of the mean value theorem is usually developed as follows: We first establish Rolle's theorem - a special case of the mean value theorem: If a function f (x) is continuous in the closed interval x1 £ x £ x2 and differentiable in the open interval x1 < x < x2, and, moreover, f(x1) = 0 and f(x1) = 0, then there exists at least one point x in the interior of the interval at which f '(x)= 0.
In fact, there must be at least one point x, interior to the interval, at which the function f(x) takes on its greatest or its least value; to be specific, we assume that x is a point where f(x) is a maximum so that for every x in the interval f(x)£f(x). Then it is certainly true for every number, the absolute value |h| of which is small enough, that f(x) - f(x + h) ³ 0 If h is positive,
we now let h tend to zero through positive values and obtain f '(x) £ 0. On the other hand, if h is negative,
and thus, by letting h tend to zero through negative values, we obtain f '(x)³ 0; comparing this result with the preceding inequality, we see that f '(x) = 0, which establishes our theorem.
We now apply Rolle's theorem to the function
which, apart from a factor independent of x, is the distance of the point (x,f(x)) of the curve from the secant, as the reader will readily verify. Obviously, this function satisfies the condition f(x) = f(x) + ax + b = 0 with constant coefficients a = -[f(x2) - f(x1)]/[x2 - x1] and b. We know already that
whence, by Rolle's theorem,
for a suitably chosen intermediate value x, and therefore
and the mean value theorem has been proved.
As the first of many applications of the mean value theorem,we shall prove the following: Let the function f(x) be continuous in the closed interval a £ x £ b and have the derivative f '(x) at every point of the open interval a < x < b. Then, if f '(x) is positive everywhere in a < x < b, the function f(x) is monotonic increasing in the interval a £ x £ b; and likewise, if f(x) is negative in a < x < b, it is monotonic decreasing.
Let f '(x) > 0 and x1, x2 > 0 be any two values of x in the closed interval. Then, by the mean value theorem,
where x1 < x < x2; Since both factors on the right hand side are positive, this proves that f(x2)>f(x1), whence f(x) is monotonic increasing.
2.3.9 The Approximate Representation of Arbitrary Functions by Linear Functions. Differentiation: The equation
which defines the derivative is equivalent to the equations
where e is a quantity which tends to zero with h = Dx. If, for the moment, we think of the point x as being fixed and the increment Dx as being variable, then, by this formula, the increment of the function, that is, the quantity Dy, consists of two terms, namely a part hf '(x), proportional to h and an error which can be made as small as we please relative to h by making h itself small enough. Thus, the smaller is the interval about the point x under consideration, the more accurately is the value of the function f(x + h) (which is a function of h), represented by its linear part f(x) + hf '(x). This approximate representation of the function f(x + h) by a linear function of h is expressed geometrically by the substitution of its tangent for the curve at the point x. In Chapter VII, we shall consider the practical application of these ideas to approximate calculations.
Here we merely remark in passing that it is possible to use this approximate representation of the increment Dy by the linear expression hf'(x) to construct a logically satisfactory definition of the notion of a differential as this was done, in particular, by Cauchy.
While the idea of the differential
as an infinitely small quantity has no meaning and it is
accordingly futile to define the derivative as the quotient of
two such quantities, we may still try to assign a sense to the
equation f '(x) = dy/dx in such a
way that the expression dy/dx need not be thought of as
purely symbolic, but as the actual quotient of two quantities dy
and dx. For this purpose, we first define the derivative
f '(x) by our limiting process, then think of x
as fixed and consider the increment h = Dx as the
independent variable. We will call this quantity h the differential
of x
and write h = dx. We now define the expression dy
= y'dx as the differential of the function y; dy is therefore a number which has nothing to
do with infinitely small quantities. Thus, 
the derivative y' = f '(x)
is now really the quotient of the differentials dy and dx;
however, there is nothing remarkable in this statement; in fact,
it is merely a restatement of the verbal definition. The
differential dy is accordingly the linear part of the
increment Dy (Fig. 16).
We shall not make any immediate use of these differentials. Nevertheless, it may be pointed out, for the sake of completeness, that we may also form second and higher differentials. In fact, if we think of h as chosen in any manner, but always the same for every value of x, then dy = hf '(x) is a function of x, of which we can again form the differential. The result will be called the second differential of y and be denoted by the symbol d²y = d²f(x). The increment of hf '(x) being h{f'(x+h) - f '(x)}, the second differential is obtained by replacing the quantity in braces by its linear part hf"(x), so that d²y == h²f"(x). We may naturally proceed further along the same lines, obtaining third, fourth, ··· differentials of y, etc., which can be defined by the expressions h3f"'(x), h4f(4)(x), etc.