Question

Use the differentiation rules developed in this section to find

the derivatives of the functions

$f\left(x\right)=2(3x+1)-4x^5$

Short answer

The derivatives of the given function is$6-20x^4$

Step-by-step solution

Step1. Given information  

Here given function is$f\left(x\right)=2(3x+1)-4x^5$

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

Step2. finding the derivatives of the function

In this question given function is $f\left(x\right)=2(3x+1)-4x^5$.

Here first we can open the parenthesis by multiplying 2 then we get

$f\left(x\right)=6x+2-4x^5$.Now we can find derivative of this function using differentiation rules , then

localid="1649266847729" $$\begin{gathered} \frac{d}{d\mathrm{x}}(6\mathrm{x}+2-4{\mathrm{x}}^5)=\frac{d\left(6\mathrm{x}\right)}{d\mathrm{x}}+\frac{d\left(2\right)}{d\mathrm{x}}-\frac{d\left(4{\mathrm{x}}^5\right)}{d\mathrm{x}} \\ \mathrm{derivative}\mathrm{of}\mathrm{a}\mathrm{constant}\mathrm{is}0,\mathrm{so}\frac{d\left(2\right)}{d\mathrm{x}}=0\mathrm{and} \\ \frac{d\left(4{\mathrm{x}}^5\right)}{d\mathrm{x}}=4\times \frac{d\left({\mathrm{x}}^5\right)}{d\mathrm{x}},\mathrm{since}\mathrm{derivative}\mathrm{of}\frac{d}{d\mathrm{x}}\left({\mathrm{x}}^{\mathrm{n}}\right)={\mathrm{nx}}^{\mathrm{n}-1},\mathrm{for}\mathrm{any}\mathrm{non}\mathrm{zero}\mathrm{rational}\mathrm{number}\mathrm{n}\mathrm{using}\mathrm{power}\mathrm{rule} \\ =4\times 5{\mathrm{x}}^4 \\ =20{\mathrm{x}}^4 \\ \frac{d\left(6\mathrm{x}\right)}{d\mathrm{x}}=6\frac{d\left(\mathrm{x}\right)}{d\mathrm{x}},\mathrm{since}\mathrm{derivative}\mathrm{of}\mathrm{an}\mathrm{identity}\mathrm{function}\mathrm{x},\frac{d\left(\mathrm{x}\right)}{d\mathrm{x}}=1 \\ =6 \\ \mathrm{then}\mathrm{the}\mathrm{derivative}\mathrm{of}\mathrm{given}\mathrm{function},\frac{d}{d\mathrm{x}}2(3\mathrm{x}+1)-4{\mathrm{x}}^5=6+0-20{\mathrm{x}}^4 \\ =6-20{\mathrm{x}}^4 \end{gathered}$$