   Chapter 3.4, Problem 17E

Chapter
Section
Textbook Problem

Find the derivative of the function.f(x) = (2x – 3)4(x2 + x + 1)5

To determine

To find:  The derivative of f(x)=(2x3)4(x2+x+1)5.

Explanation

Given:

The function is f(x)=(2x3)4(x2+x+1)5.

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)

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((2x3)4(x2+x+1)5)

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

f(x)=(2x3)4ddx((x2+x+1)5)+(x2+x+1)5ddx((2x3)4) (3)

Obtain the derivative ddx((x2+x+1)5) by using the power rule combined with the chain rule as shown equation (1).

ddx((x2+x+1)5)=5(x2+x+1)51ddx(x2+x+1)=5(x2+x+1)4(ddx(x2)+ddx(x)+ddx(1))=5(x2+x+1)4((2x21)+(1x11)+(0))=5(x2+x+1)4((2x)+(1))

=5(x2+x+1)4(2x+1)

Thus, the derivative is ddx((x2+x+1)5)=5(x2+x+1)4((2x)+(1)) (4)

Obtain the derivative ddx(2x3)4 by using the power rule combined with the chain rule as shown equation (1)

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