Math 131 Section 3 4 The Chain Rule finding the derivative of a composition Given f u 2u 4 and u x 3x 2 we know a change of 1 in x will change u x by 3 slope of u x A change of 1 in u will change f u by 2 slope of f u and in general f 2 u So a change of 1 in x changes u x by 3 which changes f u by 6 That is if x 1 then u 3 and f 2 3 6 In general f x f u u x This is the basis for the chain rule Now we will find f u x and the derivative of f with respect to x f u x 2u x 4 2 3x 2 4 6x 8 So we see the derivative of f with respect to x is 6 2 3 df The Chain Rule df du dx du dx Examples in which u x x a h x x 2 3 5 f u x u x 2 3 2 h x 5 x 4 3 2 x Here h x f u x where f u u 4 2 f u u x 5u 2 x 5 x b h x e x 2 3 c h x sin x 2 h x e 3 Examples in which a h x 1 3 4 x 4 3 2 x 2 3 1 h x 4 2 3 u x d x 1 x dx 1 3 3 1 x 2 h x 1 x 2 ex h x sin 1 x 1 x 2 2 2 3 1 x 1 1 x c h x cos x 3 2 x 2 x cos x x 1 b h x e 2 x x x 2 x 2 xe h x cos x u x 5 1 x 2 sin 1 x 2 Examples in which a h x sec b h x e 2 sec x x u x sec x u x sec x tan x h x 2 sec x sec x tan x 2 sec h x e sec x sec x tan x 2 x tan x
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