Question

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

Short 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-by-step solution

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.