# Calc03_6 - 3.6 The Chain Rule sin x 2 1 49 x2 1 sin(x 1 7 2...

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3.6 The Chain Rule

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sin x 2 + 1 sin 49 x 2 + 1 49 x 2 + 1 7 x 2 + 7 sin x 2 + 1 ( 29
g f x ( 29 ÷ g h f x ( 29 ÷ ÷ ÷ h g f x ( 29 ÷ ÷ ÷ ÷ ÷ f h h x ( 29 ÷ ÷ ÷ ÷ f g h x ( 29 ÷ ÷ ÷ ÷

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( 29 2 3 Find f (t) if f(t) = 3x - 7x ¢
( 29 ( 29 ( 29 2 3 3 3 6 4 2 5 3 Find f (t) if f(t) = 3x - 7x f (t) = 3x - 7x 3x - 7x = 9x - 42x + 49x f (t) = 54 x - 168 x + 98 x ¢ ¢

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We now have a pretty good list of “shortcuts” to find derivatives of simple functions. Of course, many of the functions that we will encounter are not so simple. What is needed is a way to combine derivative rules to evaluate more complicated functions.
Consider a simple composite function: 6 10 y x = - ( 29 2 3 5 y x = - If 3 5 u x = - then 2 y u = 6 10 y x = - 2 y u = 3 5 u x = - 6 dy dx = 2 dy du = 3 du dx = dy dy du dx du dx = × 6 2 3 = ×

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Calc03_6 - 3.6 The Chain Rule sin x 2 1 49 x2 1 sin(x 1 7 2...

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