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Techniques of Differentiation

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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: