Barrons AP Calculus

(Marvins-Underground-K-12) #1
C. THE CHAIN RULE; THE DERIVATIVE OF A COMPOSITE

FUNCTION

Formula (3) says    that

This formula is an application of the Chain Rule. For example, if we use formula
(3) to find the derivative of (x^2 − x + 2)^4 , we get


In this last equation, if we let y = ( x^2 − x + 2 )^4 and let u = x^2 − x + 2, then y =
u^4 . The preceding derivative now suggests one form of the Chain Rule:


as before. Formula (3) gives the general case where y = un and u is a
differentiable function of x.
Now suppose we think of y as the composite function f (g(x)), where y = f (u)
and u = g(x) are differentiable functions. Then


as we obtained above. The Chain Rule tells us how to differentiate the composite
function: “Find the derivative of the ‘outside’ function first, then multiply by the
derivative of the ‘inside’ one.”

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