   Chapter 3.5, Problem 57E

Chapter
Section
Textbook Problem

Find the derivative of the function. Simplify where possible.57. y = x sin − 1 x + 1 − x 2

To determine

To find: The derivative of the function.

Explanation

Given:

The function is y=xsin1x+1x2.

Derivative rules:

(1) Chain rule: dydx=dydududx

(2) Derivative of the inverse trigonometric function: ddx(sin1(x))=11x2.

Calculation:

Obtain the derivative of the function.

Consider the function y=xsin1x+1x2

Differentiate the function implicitly with respect to t,

dydx=ddx(xsin1x+1x2)=ddx(xsin1x)+ddx(1x2)

Apply the product rule (3) and simplify the expression,

dydx=xddx(sin1x)+sin1xddx(x)+ddx(1x2)=x[11x2]+sin1x(1)+ddx(1x2) [derivative rule (2)]

Let u=1x2

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