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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

Differentiate f and find the domain of f.

f ( x ) = x 1 ln ( x 1 )

To determine

To find: The derivatives of f(x) and domain of f(x).

Explanation

Given:

The function is f(x)=x1ln(x1).

Derivative Rule: Quotient Rule

If f(x). and g(x) are both differentiable function, then

ddx[fg]=gddx[f]fddx[g](g)2 (1)

Calculation:

Obtain the derivative of f(x).

f(x)=ddx(f(x))=ddx(x1ln(x1))

Apply the quotient rule as shown in equation (1),

f(x)=(1ln(x1))ddx[x](x)ddx[1ln(x1)](1ln(x1))2=(1ln(x1))[1x11](x)[01x1](1ln(x1))2=(1ln(x1))+(xx1)(1ln(x1))2=(x1)(1ln(x1))+(x)(x1)(1ln(x1))2

Simplify further and obtain the derivative,

f(x)=((x1)(x1)ln(x1))+(x)(x1)(1ln(x1))2=x<

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