   Chapter 3.4, Problem 18E

Chapter
Section
Textbook Problem

Find the derivative of the function.g(x) = (x2 + 1)3(x2 + 2)6

To determine

To find: The derivative of the function g(x)=(x2+1)3(x2+2)6.

Explanation

Given:

The function is g(x)=(x2+1)3(x2+2)6.

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 g(x).

g(x)=ddx[g(x)]=ddx[(x2+1)3(x2+2)6]

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

g(x)=(x2+1)3ddx[(x2+2)6]+(x2+2)6ddx[(x2+1)3] (3)

Obtain the derivative ddx[(x2+2)6] by using the power rule combined with the chain rule as shown equation (1).

ddx[(x2+2)6]=6(x2+2)61ddx[(x2+2)]=6(x2+2)5[ddx(x2)+ddx(2)]=6(x2+2)5[(2x21)+(0)]

Simplify further, the above derivative becomes

ddx[(x2+2)6]=6(x2+2)5[(2x)+(0)]=6(x2+2)5(2x)=12x(x2+2)5

Thus, the derivative is ddx[(x2+2)6]=12x(x2+2)5 (4)

Obtain the derivative ddx[(x2+1)3] by using the power rule combined with the chain rule as shown equation (1)

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