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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

Find f′(x). Check that your answer is reasonable by comparing the graphs of f and f′.

61. f ( x ) = 1 x 2 arcsin x

To determine

To find: The derivative of the function to check the derivative f(x), f(x) is reasonable by comparing the graph of f(x) and f(x).

Explanation

Given:

The function is f(x)=1x2arcsinx.

Derivative rules:

(1) Chain rule: dydx=dydududx

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

(3) Quotient rule: ddx(fg)=gddx(f)fddx(g)g2

Calculation:

Obtain the derivative of the function.

Consider the function f(x)=1x2arcsinx

Differentiate the function implicitly with respect to t,

f(x)=ddx(1x2arcsinx)

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

f(x)=1x2ddx(arcsinx)+arcsinxddx(1x2)=1x2(11x2)+arcsinxddx(1x2)=1+arcsinxddx(1x2)

Let v=1x2.

f(x)=1+arcsinxddx(v)

Apply the chain rule (1) and simplify the terms,

f(x)=1+arcsinxddv(v)dvdx=1+arcsinx[12v12]dvdx

Substitute v=1x2,

f(x)=1+arcsinx[12(1

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