Q.41 - Calculus

Question

Use the differentiation rules developed in this section to find

the derivatives of the functions

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

Step-by-Step Solution

Answer

The derivatives of the given function is  2x-1

Step1. Given information

 given function is $f\left(x\right)=(x+2)(x-1)$

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

Step2. finding the derivatives of the function

Here the function is in the form $f\left(x\right)=(x+2)(x-1)$.

We can find the derivatives of the given function by using the product rule.$$\begin{gathered} \mathrm{Product} \mathrm{Rule}: ( \mathrm{fg}) \left(\mathrm{x}\right) = \frac{d\mathrm{f}\left(\mathrm{x}\right)}{d\mathrm{x}} \mathrm{g}\left(\mathrm{x}\right) + \mathrm{f}\left(\mathrm{x}\right) \frac{d\mathrm{g}\left(\mathrm{x}\right)}{d\mathrm{x}} , \mathrm{where} \mathrm{f}\left(\mathrm{x}\right) \mathrm{and} \mathrm{g}\left(\mathrm{x}\right) \mathrm{are} \mathrm{the} \mathrm{functions} \mathrm{on} \mathrm{x} \\ \mathrm{So} \mathrm{in} \mathrm{this} \mathrm{question}, \mathrm{f}\left(\mathrm{x}\right)= (\mathrm{x}+2) \mathrm{and} \mathrm{g}\left(\mathrm{x}\right)=(\mathrm{x}-1) \\ \frac{d\mathrm{f}\left(\mathrm{x}\right)}{d\mathrm{x}}=\frac{d(\mathrm{x}+2)}{d\mathrm{x}} \\ =\frac{d\mathrm{x}}{d\mathrm{x}}+\frac{d\left(2\right)}{d\mathrm{x}} \\ =1 \mathrm{since} \mathrm{dervative} \mathrm{of} \mathrm{identity} \mathrm{of} \mathrm{function} \mathrm{x},\frac{d\mathrm{x}}{d\mathrm{x}}=1 . \mathrm{derivative} \mathrm{of} \mathrm{constant} \mathrm{is} \mathrm{o} \\ \frac{d\mathrm{g}\left(\mathrm{x}\right)}{d\mathrm{x}}=\frac{d(\mathrm{x}-1)}{d\mathrm{x}} \\ =\frac{d\mathrm{x}}{d\mathrm{x}}-\frac{d\left(1\right)}{d\mathrm{x}} \\ =1 \mathrm{since} \mathrm{dervative} \mathrm{of} \mathrm{identity} \mathrm{of} \mathrm{function} \mathrm{x},\frac{d\mathrm{x}}{d\mathrm{x}}=1 . \mathrm{derivative} \mathrm{of} \mathrm{constant} \mathrm{is} \mathrm{o} \\ \mathrm{When} \mathrm{we} \mathrm{applying} \mathrm{product} \mathrm{rule} \mathrm{on} \mathrm{the} \mathrm{given} \mathrm{function} \mathrm{we} \mathrm{get} \mathrm{the} \mathrm{derivative} \\ \frac{d(\mathrm{x}+2)(\mathrm{x}-1)}{d\mathrm{x}}=1\times (\mathrm{x}-1)+(\mathrm{x}+2)\times 1 \\ =\mathrm{x}-1+\mathrm{x}+2 \\ =2\mathrm{x}-1 \end{gathered}$$