#### To determine

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

#### Answer

The derivative of g(x)=(x2+1)3(x2+2)6 is 6x(x2+1)2(x2+2)5(3x2+4)_.

#### 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)]n−1g′(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)3⋅ddx[(x2+2)6]+(x2+2)6⋅ddx[(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)6−1⋅ddx[(x2+2)]=6(x2+2)5[ddx(x2)+ddx(2)]=6(x2+2)5[(2x2−1)+(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).

ddx[(x2+1)3]=3(x2+1)3−1[ddx(x2+1)]=3(x2+1)2[ddx(x2)+ddx(1)]=3(x2+1)2[(2x2−1)+(0)]

Simplify further, the above derivative becomes,

ddx[(x2+1)3]=3(x2+1)2[(2x)+(0)]=3(x2+1)2(2x)=6x(x2+1)2

Thus, the derivative ddx[(x2+1)3]=6x(x2+1)2 (5)

Substitute equations (4) and (5) in equation (3).

g′(x)=(x2+1)3[12x(x2+2)5]+(x2+2)6[6x(x2+1)2]=12x(x2+1)3(x2+2)5+6x(x2+1)2(x2+2)6=6x(x2+1)2(x2+2)5[2(x2+1)+(x2+2)]=6x(x2+1)2(x2+2)5(3x2+4)

Therefore, the derivative of g(x)=(x2+1)3(x2+2)6 is g′(x)=6x(x2+1)2(x2+2)5(3x2+4)_.