Calculus 5e_Part105

Calculus 5e_Part105 - 218 CHAPTER 3 DIFFERENTIATION RULES |...

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218 ❙❙❙❙ CHAPTER 3 DIFFERENTIATION RULES Observe that is a composite function. In fact, if we let and let , then we can write , that is, . We know how to differentiate both and , so it would be useful to have a rule that tells us how to ﬁnd the derivative of in terms of the derivatives of and . It turns out that the derivative of the composite function is the product of the deriv- atives of and . This fact is one of the most important of the differentiation rules and is called the Chain Rule. It seems plausible if we interpret derivatives as rates of change. Regard as the rate of change of with respect to , as the rate of change of with respect to , and as the rate of change of with respect to . If changes twice as fast as and changes three times as fast as , then it seems reasonable that changes six times as fast as , and so we expect that The Chain Rule If f and t are both differentiable and is the composite func- tion deﬁned by , then is differentiable and is given by the product In Leibniz notation, if and are both differentiable functions, then

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Calculus 5e_Part105 - 218 CHAPTER 3 DIFFERENTIATION RULES |...

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