Q 93 - Calculus

Question

Use implicit differentiation, the product rule, and the power rule for positive integer powers to prove the power rule for negative integer powers.

Step-by-Step Solution

Answer

$$\begin{gathered} y=x^{-n} \\ \Rightarrow y{x}^n=1 \\ \Rightarrow \frac{d}{dx}y{x}^n=\frac{d}{dx}\left(1\right) \\ \Rightarrow y\left(n{x}^{n-1}\right)+x^n\frac{dy}{dx}=0 \\ \Rightarrow n{x}^{n-1}y+x^n\frac{dy}{dx}=0 \\ \Rightarrow x^n\frac{dy}{dx}=-n{x}^{n-1}y \\ \Rightarrow \frac{dy}{dx}=-\frac{n{x}^{n-1}y}{x^n} \\ \Rightarrow \frac{dy}{dx}=-n{x}^{n-1}{x}^{-n}y \\ \Rightarrow \frac{dy}{dx}=-n{x}^{n-1-n}y \\ \Rightarrow \frac{dy}{dx}=-n{x}^{-1}y \\ \Rightarrow \frac{d}{dx}\left(x^{-n}\right)=-n{x}^{-1}{x}^{-n} \\ \Rightarrow \frac{d}{dx}\left(x^{-n}\right)=-n{x}^{-n-1} \end{gathered}$$

Hence power rule for negative integer powers proved.

Step 1. Given Information

We have to prove the power rule for negative integer powers.

That is we have to prove that :-

$\frac{d}{dx}{x}^{-n}=-n{x}^{-n-1}$.

We have to use implicit differentiation, the product rule, and the power rule for positive integer powers to prove this thing.

Step 2. Prove the power rule for negative integer powers

Consider the following function :-

$y=x^{-n}$

We can write it as :-

$y{x}^n=1$

Take differentiation on both sides, the we have :-

$\frac{d}{dx}y{x}^n=\frac{d}{dx}\left(1\right)$

Apply the  the product rule, and the power rule for positive integer powers, then we have :-

$$\begin{gathered} y\left(n{x}^{n-1}\right)+x^n\frac{dy}{dx}=0 \\ \Rightarrow n{x}^{n-1}y+x^n\frac{dy}{dx}=0 \\ \Rightarrow x^n\frac{dy}{dx}=-n{x}^{n-1}y \\ \Rightarrow \frac{dy}{dx}=-\frac{n{x}^{n-1}y}{x^n} \\ \Rightarrow \frac{dy}{dx}=-n{x}^{n-1}{x}^{-n}y \\ \Rightarrow \frac{dy}{dx}=-n{x}^{n-1-n}y \\ \Rightarrow \frac{dy}{dx}=-n{x}^{-1}y \end{gathered}$$

Put the value $y=x^{-n}$, then we have :-

$$\begin{gathered} \frac{d}{dx}\left(x^{-n}\right)=-n{x}^{-1}{x}^{-n} \\ \Rightarrow \frac{d}{dx}\left(x^{-n}\right)=-n{x}^{-n-1} \end{gathered}$$

This is the required value.

Hence power rule for negative integer powers proved.