# Whether the statement, “If f is differentiable, then d d x [ f ( x ) ] = f ′ ( x ) 2 x ” is true or false.

### 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, Problem 5RQ
To determine

## Whether the statement, “If f is differentiable, then ddx[f(x)]=f′(x)2x” is true or false.

Expert Solution

The statement is false.

### Explanation of Solution

Result used: Chain Rule

If g is differentiable at t and f is differentiable at g(t), then the composite function F=fg defined by F(t)=f(g(t)) is differentiable at x and F is given by the product

F(t)=f(g(t))g(t) (1)

Calculation:

Consider ddx[f(x)]=f(x)2x.

Take the left hand side ddx[f(x)].

By the chain rule,

ddx[f(x)]=dd(x)[f(x)]d(x)dx=f(x)12x=f(x)2x

Since the left hand side is not equal to the right side. That is, ddx[f(x)]f(x)2x.

Counterexample:

The function f(x)=sin(x).

By the chain rule,

f(x)=cos(x)2xcosx2x=f(x)2x

Hence, the given statement is false.

### Have a homework question?

Subscribe to bartleby learn! Ask subject matter experts 30 homework questions each month. Plus, you’ll have access to millions of step-by-step textbook answers!