Q.29 - Calculus

Question

Differentiate each of the functions in Exercises 29–34 in two

different ways: first with the product and/or quotient rules and

then without these rules. Then use algebra to show that your

answers are the same.

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

Step-by-Step Solution

Answer

The Derivative of the given function is $3x^2+2x$

Step1. Given information

Given function is $f\left(x\right)=x^2(x+1)$.

We have to differentiate first with the product rule and

then without these rules, after that, we have to use algebra to show that  the answers are the same.

Step 2. Differentiate using the product rule

The given function is $f\left(x\right)=x^2(x+1)$.

The  $\mathrm{Product} \mathrm{Rule}: ( {\mathrm{f}}^{\text{'}}\mathrm{g}) \left(\mathrm{x}\right) = \mathrm{f}\text{'} \left(\mathrm{x}\right) \mathrm{g}\left(\mathrm{x}\right) + \mathrm{f}\left(\mathrm{x}\right) \mathrm{g} \text{'}\left(\mathrm{x}\right)$

When we applying product rule on the given function we get,

$$\begin{gathered} \mathrm{Product} \mathrm{Rule}: ( {\mathrm{f}}^{\text{'}}\mathrm{g}) \left(\mathrm{x}\right) = \mathrm{f}\text{'} \left(\mathrm{x}\right) \mathrm{g}\left(\mathrm{x}\right) + \mathrm{f}\left(\mathrm{x}\right) \mathrm{g} \text{'}\left(\mathrm{x}\right) \\ \mathrm{here} \mathrm{take} \mathrm{f}\left(\mathrm{x}\right)={\mathrm{x}}^2 \mathrm{and} \mathrm{g}\left(\mathrm{x}\right)=(\mathrm{x}+1) \\ \mathrm{f}\text{'}\left(\mathrm{x}\right)=\frac{\mathrm{d}\left({\mathrm{x}}^2\right)}{\mathrm{dx}} \\ =2\mathrm{x} , \mathrm{since} \mathrm{based} \mathrm{on} \mathrm{power} \mathrm{rule} \frac{\mathrm{d}\left({\mathrm{x}}^{\mathrm{n}}\right)}{\mathrm{dx}}={\mathrm{nx}}^{\mathrm{n}} \\ \mathrm{g}\text{'}\left(\mathrm{x}\right)=\frac{\mathrm{d}(\mathrm{x}+1)}{\mathrm{dx}} \\ =\frac{\mathrm{d}\left(\mathrm{x}\right)}{\mathrm{dx}}+\frac{\mathrm{d}\left(1\right)}{\mathrm{dx}} \\ =1 \\ \mathrm{Then} \mathrm{based} \mathrm{on} \mathrm{product} \mathrm{rule} \mathrm{derivative} \mathrm{of} \mathrm{the} \mathrm{function} \\ \mathrm{f}\frac{d\left({\mathrm{x}}^2\right(\mathrm{x}+1)}{d\mathrm{x}}= 2\mathrm{x}(\mathrm{x}+1)+{\mathrm{x}}^2\left(1\right) \\ =2{\mathrm{x}}^2+2\mathrm{x}+{\mathrm{x}}^2 \end{gathered}$$

Step 3. Differentiate with out using product rule

In this function  first, we have to use the distributive property, so  multiply x² by the terms inside the parenthesis, then we get 

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right)={\mathrm{x}}^2(\mathrm{x}+1) \\ \mathrm{f}\left(\mathrm{x}\right)={\mathrm{x}}^3+{\mathrm{x}}^2 \\ \mathrm{now} \mathrm{we} \mathrm{can} \mathrm{find} \mathrm{the} \mathrm{derivative} \mathrm{using} \mathrm{power} \mathrm{rule} ,\mathrm{for} \mathrm{any} \mathrm{non} \mathrm{rational} \mathrm{integer} \mathrm{n}, \frac{d{\mathrm{x}}^{\mathrm{n}}}{d\mathrm{x}}={\mathrm{nx}}^{\mathrm{n}-1} \\ \frac{d({\mathrm{x}}^3+{\mathrm{x}}^2)}{d\mathrm{x}}=\frac{d\left({\mathrm{x}}^3\right)}{d\mathrm{x}}+\frac{d\left({\mathrm{x}}^2\right)}{d\mathrm{x}} \\ =3{\mathrm{x}}^2+2\mathrm{x} \\ \mathrm{So} \mathrm{derivative} \mathrm{of} \mathrm{the} \mathrm{given} \mathrm{function} \mathrm{is} 3{\mathrm{x}}^2+2\mathrm{x} \end{gathered}$$

Step 4. Checking the answers are same

Here  without using the product rule  the derivative of a given function $\mathrm{f}\left(\mathrm{x}\right)={\mathrm{x}}^2(\mathrm{x}+1) \mathrm{is} 3{\mathrm{x}}^2+2\mathrm{x}$

The derivative of the function using the product rule is  $$\begin{gathered} \frac{d\left({\mathrm{x}}^2\right(\mathrm{x}+1\left)\right)}{d\mathrm{x}}=2{\mathrm{x}}^2+2\mathrm{x}+{\mathrm{x}}^2 \\ \mathrm{When} \mathrm{we} \mathrm{using} \mathrm{algebra} , \mathrm{combining} \mathrm{the} \mathrm{like} \mathrm{term} \mathrm{and} \mathrm{adding} \mathrm{togethor} \mathrm{we} \mathrm{get} \\ \frac{d\left({\mathrm{x}}^2\right(\mathrm{x}+1\left)\right)}{d\mathrm{x}}=2{\mathrm{x}}^2+2\mathrm{x}+{\mathrm{x}}^2 \\ =2{\mathrm{x}}^2 +{\mathrm{x}}^2+2\mathrm{x} \\ =3{\mathrm{x}}^2+2\mathrm{x} \end{gathered}$$

 thus in both cases answers are same