BuyFind

Single Variable Calculus: Concepts...

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

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

Answer to Problem 5RQ

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.

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