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.

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

Short answer

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

Step-by-step solution

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 localid="1648743667452" $\mathrm{Product}\mathrm{Rule}:({\mathrm{f}}^{\text{'}}\mathrm{g})\left(\mathrm{x}\right)=\mathrm{f}\text{'}\left(\mathrm{x}\right)\mathrm{g}\left(\mathrm{x}\right)+\mathrm{f}\left(\mathrm{x}\right)\mathrm{g}\text{'}\left(\mathrm{x}\right)$

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

localid="1649319615141" $$\begin{gathered} \mathrm{Product}\mathrm{Rule}:({\mathrm{f}}^{\text{'}}\mathrm{g})\left(\mathrm{x}\right)=\mathrm{f}\text{'}\left(\mathrm{x}\right)\mathrm{g}\left(\mathrm{x}\right)+\mathrm{f}\left(\mathrm{x}\right)\mathrm{g}\text{'}\left(\mathrm{x}\right) \\ \mathrm{here}\mathrm{take}\mathrm{f}\left(\mathrm{x}\right)={\mathrm{x}}^2\mathrm{and}\mathrm{g}\left(\mathrm{x}\right)=(\mathrm{x}+1) \\ \mathrm{f}\text{'}\left(\mathrm{x}\right)=\frac{\mathrm{d}\left({\mathrm{x}}^2\right)}{\mathrm{dx}} \\ =2\mathrm{x},\mathrm{since}\mathrm{based}\mathrm{on}\mathrm{power}\mathrm{rule}\frac{\mathrm{d}\left({\mathrm{x}}^{\mathrm{n}}\right)}{\mathrm{dx}}={\mathrm{nx}}^{\mathrm{n}} \\ \mathrm{g}\text{'}\left(\mathrm{x}\right)=\frac{\mathrm{d}(\mathrm{x}+1)}{\mathrm{dx}} \\ =\frac{\mathrm{d}\left(\mathrm{x}\right)}{\mathrm{dx}}+\frac{\mathrm{d}\left(1\right)}{\mathrm{dx}} \\ =1 \\ \mathrm{Then}\mathrm{based}\mathrm{on}\mathrm{product}\mathrm{rule}\mathrm{derivative}\mathrm{of}\mathrm{the}\mathrm{function} \\ \mathrm{f}\frac{d\left({\mathrm{x}}^2\right(\mathrm{x}+1)}{d\mathrm{x}}=2\mathrm{x}(\mathrm{x}+1)+{\mathrm{x}}^2\left(1\right) \\ =2{\mathrm{x}}^2+2\mathrm{x}+{\mathrm{x}}^2 \end{gathered}$$

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