# The differentiation of the function y = 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 3.2, Problem 5E
To determine

## To find: The differentiation of the function y=xex.

Expert Solution

The differentiation of the function y=xex is dydx=1xex.

### Explanation of Solution

Given:

The function y=xex.

Derivative rules:

(1) Quotient Rule: If f(x) and g(x) are both differentiable, then

ddx[f(x)g(x)]=g(x)ddx[f(x)]f(x)ddx[g(x)][g(x)]2

(2) Power rule: ddx(xn)=nxn1

(3) Derivative of exponential function: ddx(ex)=ex

Calculation:

The derivative of the function is dydx, which is obtained as follows,

dydx=ddx(xex)

Substitute x for f(x) and ex for g(x) in the quotient rule (1),

dydx=exddx[x]xddx[ex][ex]2

Apply the derivative rule (3) and the power rule (2),

dydx=ex(1x11)x(ex)exex=exxexexex=ex(1x)exex=1xex

Therefore, the differentiation of the function y=xex is dydx=1xex.

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