# To show that g ' ( x ) = x f ' ( x ) + f ( x ) using the definition of derivative. ### 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 2, Problem 16P
To determine

## To show that g'(x)=xf'(x)+f(x) using the definition of derivative.

Expert Solution

### Explanation of Solution

Given:

The f is a differentiable function and g(x)=xf(x)

Calculation:

Since f is a differentiable function, the derivative exist.

So by the definition of derivative-

f'(x)=limh0(x+h)f(x+h)xf(x)h=limh0xf(x+h)+hf(x+h)xf(x)h=limh0xf(x+h)xf(x)+hf(x+h)h=limh0xf(x+h)xf(x)h+limh0hf(x+h)h

It is known that,

f'(x) = limh0 [f(x + h) - g(x)] / h

Therefore, on substituting

f'(x)=x f'(x) +limh0 [ h f(x + h) / h ]= x f '(x) + f(x)

Hence proved.

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