This fact is one of the most important of the

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Unformatted text preview: f f and t. 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 du d x as the rate of change of u with respect to x, dy du as the rate of change of y with respect to u, and dy d x as the rate of change of y with respect to x. If u changes twice as fast as x and y changes three times as fast as u, then it seems reasonable that y changes six times as fast as x, and so we expect that dy dx dy du du dx The Chain Rule If f and t are both differentiable and F f t is the composite function defined by F x f t x , then F is differentiable and F is given by the product Fx In Leibniz notation, if y f tx t x f u and u dy dx Comments on the Proof of the Chain Rule Let t x are both differentiable functions, then dy du du dx u be the change in u corresponding to a change of x in x, that is, u tx x tx fu u fu Then the corresponding change in y is y It is tempting to wr...
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