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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.
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
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) |
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) = = = |
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:
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