Question

Use the differentiation rules developed in this section to find

the derivatives of the functions

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

Short answer

The derivative of the given function is$12x$

Step-by-step solution

Step1. Given information

The function $f\left(x\right)=2(1+3x^2)$

We have to find out the derivatives of the given function.

Step2. finding the derivatives of the function

Based on the question

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right)=2(1+3{\mathrm{x}}^2),\mathrm{we}\mathrm{can}\mathrm{open}\mathrm{the}\mathrm{parenthesis}\mathrm{by}\mathrm{multipying}2\mathrm{we}\mathrm{get}, \\ =2+6{\mathrm{x}}^2, \end{gathered}$$

Now using the differentiation rule to find the derivate of the given function , we get

$$\begin{gathered} \frac{d}{d\mathrm{x}}(2+6{\mathrm{x}}^2)=\frac{d\left(2\right)}{d\mathrm{x}}+\frac{d\left(6{\mathrm{x}}^2\right)}{d\mathrm{x}} \\ \mathrm{derivative}\mathrm{of}\mathrm{a}\mathrm{constant}\mathrm{is}0,\mathrm{so}\frac{d\left(2\right)}{d\mathrm{x}}=0\mathrm{and} \\ \frac{d\left(6{\mathrm{x}}^2\right)}{d\mathrm{x}}=6\frac{d\left({\mathrm{x}}^2\right)}{d\mathrm{x}},\sin ce\mathrm{derivative}\mathrm{of}\frac{d}{d\mathrm{x}}\left({\mathrm{x}}^{\mathrm{n}}\right)=n{x}^{\mathrm{n}-1},\mathrm{for}\mathrm{any}\mathrm{non}\mathrm{zero}\mathrm{rational}\mathrm{number}\mathrm{n}\mathrm{using}\mathrm{power}\mathrm{rule} \\ =6\times 2x \\ =12x \\ \mathrm{th}en\mathrm{the}\mathrm{derivative}\mathrm{of}\mathrm{given}\mathrm{function},\frac{d}{d\mathrm{x}}2(1+x^2)=0+12x \\ =12x \end{gathered}$$