   Chapter 3.4, Problem 57E

Chapter
Section
Textbook Problem

(a) If f ( x ) = 2 − x 2 x , find f′(x).(b) Check to see that your answer to part (a) is reasonable by comparing the graphs of f and f′.

(a)

To determine

To find: The derivative of f(x)=x2x2.

Explanation

Given:

The function is f(x)=x2x2.

Result used:

The Power Rule combined with the Chain Rule:

If n is any real number and g(x) is differentiable function, then

ddx[g(x)]n=n[g(x)]n1g(x) (1)

Derivative Rule: Product Rule:

If f(x). and g(x) are both differentiable function, then

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

Calculation:

Obtain the derivative of f(x).

f(x)=ddx(f(x))=ddx(x2x2)

Apply the product rule as shown in equation (2),

f(x)=xddx[2x2]+2x2ddx[x]=xddx(2x2)12+2x2[1x11]=xddx(2x2)12+2x2[x0]=xddx(2x2)12+2x2

(b)

To determine

To sketch: The graph of f(x) and f(x).

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