   Chapter 3.11, Problem 43E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# Find the derivative. Simplify where possible.43. y = x   sinh − ( x / 3 ) − 9 + x 2

To determine

To find: The derivative of the function.

Explanation

Given:

The function y=xsinh1(x3)9+x2.

Derivative rules used:

(1) The chain rule: dydx=dydududx

(2) The derivative of the inverse hyperbolic sine function is, ddx(sinh1x)=11+x2.

Calculation:

The derivative of the function y=xsinh1(x3)9+x2 is computed as follows,

dydx=ddx(xsinh1(x3)9+x2)=ddx(xsinh1(x3))ddx(9+x2)=[xddx(sinh1(x3))+sinh1(x3)ddx(x)]ddx(9+x2)=[x11+(x3)2ddx(x3)+sinh1(x3)(1)]ddx(9+x2)

Let u=9+x2 and apply the chain rule,

dydx=[x11+x29(13)+sinh1(x3)]ddx(u

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