   Chapter 3.5, Problem 54E ### 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 of the function. Simplify where possible.54. y = tan − 1 ( x − 1 + x 2 )

To determine

To find: The derivative of the function.

Explanation

Given:

The function is y=tan1(x1+x2).

Derivative rules:

(1) Chain rule: If y=f(u) and u=g(x)  are both differentiable function, then

dydx=dydududx.

(2) Derivative of the inverse trigonometric function: ddx(tan1(x))=11+x2.

Calculation:

Obtain the derivative of the function.

Consider the function y=tan1(x1+x2)

Differentiate the function with respect to x,

dydx=ddx(tan1(x1+x2))

Let .

dydx=ddx(tan1(u))

Apply the chain rule (1),

dydx=ddu(tan1(u))dudx    dydx=(11+u2)dudx            [by derivative rule(2)]

Substitute u=x1+x2,

dydx=(11+(x1+x2)2)ddx(x1+x2)=(11+x2+(1+x2)22x1+x2)(1ddx(1+x2)12)=(11+x2+1+x22x1+x2)(1ddx(1+x2)12)

Let v=1+x2

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