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
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
In Leibniz notation, if y f tx t x f u and u
dx Comments on the Proof of the Chain Rule Let t x are both differentiable functions, then
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
It is tempting to wr...
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