Differential and Integral Calculus

3.1 The Simplest Rules For Differentiation and their Applications		3.7 Some Applications of the Exponential Function
3.1.1 Rules for Differentiation		3.7.1 Definition of the Exponential Function by Means of a Differential Equation
3.1.2 Differentiation of the Rational Functions		3.7.2 Interest Compounded Continuously. Radio-active Disintegration
3.1.3 Differentiation of the Trigonometric Functions		3.7.3 Cooling or Heating of a Body by a Surrounding Medium
3.2 The corresponding Integral formulae		3.7.4 Variation of the Atmospheric Pressure with the Height above the Surface of the Earth
3.2.1. General Rules for Integration		3.7.5 Progress of a Chemical Solution
3.2.2 Integration of the Simplest Functions		3.7.6 Making and Breaking of an Electric Circuit
3.3 The Inverse Function and its derivative		3.8 The Hyperbolic Functions
3.3.1 The General Formula for Differentiation		3.8.1 Analytical Definition
3.3.2 The Inverse of the Power Function		3.8.2 Addition Theorems and Formulae for Differentiation
3.3.3 The Inverse Trigonometric Functions:		3.8.3 The Inverse Hyperbolic Functions
3.3.4 The Corresponding Integral Formulae		3.8.4 Further Analogies
3.4 Differentiation of a function of a function		3.9 The Order of Magnitude of Funtions
3.4.1 The Chain Rule		3.9.1 The Concept of Order of Magnitude. The Simplest Cases
3.4.2 Examples		3.9.2 The Order of Magnitude of the Exponential Function and the Logarithm
3.4.3 Further Remarks on the Integration and Differentiation of x^a, when a is Irrational		3.9.3 General Remarks
3.5 Maxima and Minima		3.9.4 The Order of Magnitude of a Function in the Neighbourbood of an Arbitrary Point
3.5.1 Convexity or Concavity of Curve		3.9.5 Tbe Order of Magnitude of a Function tending to Zero
3.5.2 Maxima and Minima		Appendix to Chapter III
3.5.3 Examples of Maxima and Minima		A3.1 Some Special Functions
3.6 The Logarithm and the Exponential function		A3.1.1 The Function y = e^-1/x²
3.6.1 Definition of the Logarithm. The Differentiation Formula		A3.1.2. The Function y = e^-1/x
3.6.2 The Addition Theorem		A3.1.3 The function y = tanh 1/x
3.6.3. Monotonic Character and Values of the Logarithm		A3.1.4 The Function y =x tanh 1/x
3.6.4 The Inverse Function of the Logarithm (the Exponential Function)		A3.1.5 The Function y = x sin 1/x, y(0) = 0
3.6.5 The General Exponential Function a^x and the General Power x^a		A3.2 Remarks on the Differentiability of Functions
3.6.6. The Exponential function and the Logaritlm represented as Limits
3.6.7 Final Remarks