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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 the 1000th derivative of f(x) = xex.

To determine

To find: The 1000th derivative of f(x)=xex.

Explanation

Given:

The function is f(x)=xex.

Derivative Rule: Product Rule

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

ddx[f(x)g(x)]=f(x)ddx[g(x)]+g(x)ddx[f(x)] (1)

Calculation:

The first derivative f(x)=xex is computed as follows,

f(x)=ddx(f(x))=ddx(xex)

Apply product rule as shown in equation (1),

f(x)=xddx(ex)+exddx(x)=x(ex)+ex(1)=(1x)ex

The second derivative of f(x) is computed as follows,

f(x)=ddx(f(x))=ddx((1x)ex)

Apply product rule as shown in equation (1),

f(x)=(1x)ddx(ex)+exddx(1x)=(1x)(ex)+ex(01)=(1x)(e−<

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