13142109-17-the-Chain-Rule

# 13142109-17-the-Chain-Rule - Math 135 Class Notes Business...

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Math 135 Business Calculus Spring 2009 Class Notes 1.7 The Chain Rule THE EXTENDED POWER RULE According to the Power Rule, the derivative of the power function y = x k is given by d dx ( x k ) = kx k 1 . For a power function y = x k , we have a variable x raised to a power k . What is the derivative if, instead of the variable x raised to a power, we have a function raised to a power, such as d dx £ g ( x ) § k In particular, suppose we want the derivative of y = £ g ( x ) § 2 . We can compute this using the Product Rule: d dx £ g ( x ) § 2 = d dx £ g ( x ) · g ( x ) § = g 0 ( x ) · g ( x ) + g ( x ) · g 0 ( x ) = 2 g ( x ) · g 0 ( x ) This resembles the result of di±erentiating y = x 2 . The exponent 2 has been pulled down in front of the function and the power has been reduced by 1. However, we also have a factor of g 0 ( x ) that appears in the derivative. This is a special case of the following general rule: THEOREM 7 The Extended Power Rule Suppose that g ( x ) is a di±erentiable function of x . Then, for any real number k , d dx £ g ( x ) § k

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13142109-17-the-Chain-Rule - Math 135 Class Notes Business...

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