Q.36 - Calculus

Question

Use the differentiation rules developed in this section to find

the derivatives of the functions

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

Step-by-Step Solution

Answer

The derivative of $\mathrm{f}\left(\mathrm{x}\right) \mathrm{is} 2+8\mathrm{x}$

Step1. Given information

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

We need to find out the the derivatives of the functions

Step2. finding the derivatives of the function

When we differentiate  the given function, by using differentiation rules  we get 

$$\begin{gathered} \frac{d}{d\mathrm{x}}(2\mathrm{x}-3+4{\mathrm{x}}^2)=\frac{d\left(2\mathrm{x}\right)}{d\mathrm{x}}-\frac{d\left(3\right)}{d\mathrm{x}}+\frac{d\left(4{\mathrm{x}}^2\right)}{d\mathrm{x}} \\ \mathrm{since} \mathrm{derivative} \mathrm{of} \mathrm{a} \mathrm{constant} \mathrm{is} 0 , \mathrm{so}\frac{d\left(3\right)}{d\mathrm{x}}=0 \mathrm{and} \\ \mathrm{derivative} \mathrm{of} \frac{d}{d\mathrm{x}}\left({\mathrm{x}}^{\mathrm{n}}\right)={\mathrm{nx}}^{\mathrm{n}-1} ,\mathrm{for} \mathrm{any} \mathrm{non} \mathrm{zero} \mathrm{rational} \mathrm{number} \mathrm{n} \mathrm{using} \mathrm{power} \mathrm{rule} \mathrm{then} \frac{d\left(4{\mathrm{x}}^2\right)}{d\mathrm{x}}=4\frac{d\left({\mathrm{x}}^2\right)}{d\mathrm{x}} \\ =4\times 2\mathrm{x} \\ = 8\mathrm{x} \\ \mathrm{we} \mathrm{know} \mathrm{that} \mathrm{derivetive} \mathrm{of} \mathrm{identity} \frac{d\left(\mathrm{x}\right)}{d\mathrm{x}}=1 ,\mathrm{so} \frac{d\left(2\mathrm{x}\right)}{d\mathrm{x}}=2\frac{d\left(\mathrm{x}\right)}{d\mathrm{x}} \\ =2 \\ \mathrm{thus} \mathrm{the} \mathrm{derivative} \mathrm{of} \mathrm{given} \mathrm{function} ,\frac{d}{d\mathrm{x}}(2\mathrm{x}-3+4{\mathrm{x}}^{2 }) =2-0+8\mathrm{x} \\ =2-8\mathrm{x} \end{gathered}$$