Q.37 - Calculus

Question

Use the differentiation rules developed in this section to find

the derivatives of the functions

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

Step-by-Step Solution

Answer

The derivative of  the given  function is $12x$

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

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