Q.39 - Calculus

Question

Use the differentiation rules developed in this section to find

the derivatives of the functions

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

Step-by-Step Solution

Answer

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

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

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