# The derivative of the function.

### 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 47E

(a)

To determine

## To find: The derivative of the function.

Expert Solution

The derivative of the function y=xg(x) is dydx=xg(x)+g(x).

### Explanation of Solution

Given:

The function y=xg(x).

Derivative rule:

(1) Product Rule: ddx[f1(x)f2(x)]=f1(x)ddx[f2(x)]+f2(x)ddx[f1(x)]

(2) Quotient Rule: If f1(x) and f2(x) are both differentiable, then

ddx[f1(x)f2(x)]=f2(x)ddx[f1(x)]f1(x)ddx[f2(x)][f2x]2

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

Calculation:

The derivative of the function y=xg(x) is dydx , which is obtained as follows,

dydx=ddx(xg(x))

Apply the product rule (1) and the power rule (3),

dydx=ddx(xg(x))=xddx(g(x))+g(x)ddx(x)=xg(x)+g(x)(1)=xg(x)+g(x)

Therefore, the derivative of the function y=xg(x) is dydx=xg(x)+g(x).

(b)

To determine

### To find: The derivative of the function.

Expert Solution

The derivative of the function y=xg(x) is dydx=g(x)xg(x)[g(x)]2.

### Explanation of Solution

Given:

The function y=xg(x).

Calculation:

The derivative of the function y=xg(x) is dydx , which is obtained as follows,

dydx=ddx(xg(x))

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

dydx=g(x)ddx(x)xddx(g(x))[g(x)]2=g(x)(1)xg(x)[g(x)]2=g(x)xg(x)[g(x)]2

Therefore, the derivative of the function y=xg(x) is dydx=g(x)xg(x)[g(x)]2.

(c)

To determine

### To find: The derivative of the function.

Expert Solution

The derivative of the function y=g(x)x is dydx=xg(x)g(x)x2.

### Explanation of Solution

Given:

The function y=g(x)x

Calculation:

The derivative of the function y=g(x)x is dydx , which is obtained as follows,

dydx=ddx(g(x)x)

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

dydx=xddx(g(x))g(x)ddx(x)[x]2=xg(x)g(x)(1)x2=xg(x)g(x)x2

Therefore, the derivative of the function y=g(x)x is dydx=xg(x)g(x)x2.

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