Question

Find the derivatives of the functions in Exercises 21–46. Keep in mind that it may be convenient to do some preliminary algebra before differentiating.

$f\left(x\right)=x{(3x^2+1)}^9$

Short answer

The required answer is${(3x^2+1)}^9+54x^2{(3x^2+1)}^8$

Step-by-step solution

Step 1. Given Information 

The given function is$f\left(x\right)=x{(3x^2+1)}^9$

Step 2. Calculation 

Differentiate both the sides with respect to x, we get,

$$\begin{gathered} f\text{'}\left(x\right)=\frac{d}{dx}\left(x\right){(3x^2+1)}^9+x\left(\frac{d}{dx}{(3x^2+1)}^9\right) \\ ={(3x^2+1)}^9+x\left(9{(3x^2+1)}^{9-1}\frac{d}{dx}(3x^2+1)\right) \\ ={(3x^2+1)}^9+x\left(9{(3x^2+1)}^8\left(\frac{d}{dx}3x^2+\frac{d}{dx}1\right)\right) \\ ={(3x^2+1)}^9+x\left(9{(3x^2+1)}^8\left(3\left(2x\right)\right)\right) \\ ={(3x^2+1)}^9+54x^2{(3x^2+1)}^8 \end{gathered}$$