Q 89 - Calculus

Question

In Exercise $89$ of the Section $2.3$ you used the definition of derivative to prove the quotient rule. Prove it now another way: by writing a quotient $\frac{f}{g}$ as a product and applying the product, power, and chain rules. Point out where you use each rule.

Step-by-Step Solution

Answer

$$\begin{gathered} y=\frac{f}{g} \\ \Rightarrow yg=f \\ \Rightarrow \frac{d}{dx}\left(yg\right)=\frac{d}{dx}\left(f\right) \\ \Rightarrow y\frac{dg}{dx}+g\frac{dy}{dx}=\frac{df}{dx} \\ \Rightarrow g\frac{dy}{dx}=\frac{df}{dx}-y\frac{dg}{dx} \\ \Rightarrow \frac{dy}{dx}=\frac{\frac{df}{dx}-y\frac{dg}{dx}}{g} \\ \Rightarrow \frac{d}{dx}\left(\frac{f}{g}\right)=\frac{{\displaystyle \frac{df}{dx}-\frac{f}{g}\frac{dg}{dx}}}{g} \\ \Rightarrow \frac{d}{dx}\left(\frac{f}{g}\right)=\frac{g\frac{df}{dx}-f\frac{dg}{dx}}{g^2} \end{gathered}$$

Hence proved.

Step 1. Given Information

We have to prove the quotient rule of derivative.

That is we have to prove that :-

$\frac{d}{dx}\left(\frac{f}{g}\right)=\frac{g{\displaystyle \frac{df}{dx}}-f{\displaystyle \frac{dg}{dx}}}{g^2}$

We have to convert quotient to a product. Then we have to use product, quotient and chain rule to prove this rule.

Step 2. Proof of quotient rule

Consider the following quotient :-

$$\begin{gathered} y=\frac{f}{g} \\ \Rightarrow yg=f \end{gathered}$$

Take derivative on both sides, then we have :-

$\frac{d}{dx}\left(yg\right)=\frac{d}{dx}\left(f\right)$

Now use product and chain rule, then we have :-

$$\begin{gathered} y\frac{dg}{dx}+g\frac{dy}{dx}=\frac{df}{dx} \\ \Rightarrow g\frac{dy}{dx}=\frac{df}{dx}-y\frac{dg}{dx} \\ \Rightarrow \frac{dy}{dx}=\frac{\frac{df}{dx}-y\frac{dg}{dx}}{g} \end{gathered}$$

Put the value of $y=\frac{f}{g}$, then we have :-

$$\begin{gathered} \frac{d}{dx}\left(\frac{f}{g}\right)=\frac{{\displaystyle \frac{df}{dx}}-{\displaystyle \frac{f}{g}}{\displaystyle \frac{dg}{dx}}}{g} \\ \Rightarrow \frac{d}{dx}\left(\frac{f}{g}\right)=\frac{g\frac{df}{dx}-f\frac{dg}{dx}}{g^2} \end{gathered}$$

This is the required result.

Hence the quotient rule is proved.