In this section, we introduce the basic techniques of differentiation and apply them to functions built up from the elementary functions.

### Basic Properties of Differentiation

There are two simple properties of differentiation that make the calculation of derivatives much easier. Let f (x), g(x) be two functions, and let c be a constant. Then

1. [cf (x)] = cf'(x)
2. (f + g)'(x) = f'(x) + g'(x)
In words, these properties say that the derivative of a constant times a function is that constant times the derivative of the function, and the derivative of a sum of functions is the sum of the derivatives of the functions.

### Product Rule

Given two functions f (x), g(x), and their derivatives f'(x), g'(x), we would like to be able to calculate the derivative of the product function f (x)g(x). We do this by follwowing the product rule: [f (x)g(x)] =  =  + = f (x + ε)  g(x) = f (x)g'(x) + g(x)f'(x)

### Quotient Rule

Now we show how to express the derivative of the quotient of two functions f (x), g(x) in terms of their derivatives f'(x), g'(x). Let q(x) = f (x)/g(x). Then f (x) = q(x)g(x), so by the product rule, f'(x) = q(x)g'(x) + g(x)q'(x). Solving for q'(x), we obtain

 q'(x) = = = This is known as the quotient rule. As an example of the use of the quotient rule, consider the rational function q(x) = x/(x + 1). Here f (x) = x and g(x) = x + 1, so

 q'(x) = = = ### Chain Rule

Suppose a function h is a composition of two other functions, that is, h(x) = f (g(x)). We would like to express the derivative of h in terms of the derivatives of f and g. To do so, follow the chain rule, given below: