## Calculus (3rd Edition)

Published by W. H. Freeman

# Chapter 3 - Differentiation - 3.7 The Chain Rule - Exercises - Page 146: 44

#### Answer

y'=2(x+2)cos($x^{2}$+4x)

#### Work Step by Step

The chain rule states that if y=g(h(x)), then y'=g'(h(x))h'(x). Since y=sin($x^{2}$+4x), we can set g(x)=sin(x) (outside function) and h(x)=$x^{2}$+4x (inside function), and use the chain rule to find y'. Therefore, y'=g'($x^{2}$+4x)h'(x) We know that $\frac{d}{dx}$[sin(x)]=cos(x), which is a derivative that would be a good idea to have memorized $\frac{d}{dx}$[$x^{2}$+4x]=2x+4, using the power rule Now that we know that g'(x)=cos(x) and h'(x)=2x+4, we can find y'. y'=cos($x^{2}$+4x)*(2x+4) =2(x+2)cos($x^{2}$+4x)

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