# The 1000th derivative of f ( x ) = x e − x . ### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805 ### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

#### Solutions

Chapter 3.4, Problem 68E
To determine

## To find: The 1000th derivative of f(x)=xe−x.

Expert Solution

The 1000th derivative of f(x)=xex is f1000(x)=(x1000)ex_.

### Explanation of Solution

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)(ex)ex=(x11)ex

=(x2)ex

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

f(x)=ddx(f(x))=ddx((x2)ex)

Apply product rule as shown in equation (1),

f(x)=(x2)ddx(ex)+exddx(x2)=(x2)(ex)+ex(10)=(x+2)(ex)+ex=(2x+1)ex

=(3x)ex

Proceed in the similar way, it is noticed that the nth derivative is expressed as follows,

fn(x)={(xn)exif n is even(nx)exif n is odd

Since n=1000 is even, the derivative is f1000(x)=(x1000)ex.

Therefore, the 1000th derivative of f(x)=xex is f1000(x)=(x1000)ex_.

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