Question

given the function $y = \frac{1}{x^{3}}$, find $\frac{dy}{dx}$. express your answer in simplest form without using negative exponents.

Explanation:

Step1: Rewrite the function

Rewrite $y = \frac{1}{x^{3}}$ as $y=x^{-3}$ using the negative - exponent rule $\frac{1}{a^{n}}=a^{-n}$.

Step2: Apply the power rule

The power rule for differentiation is $\frac{d}{dx}(x^{n})=nx^{n - 1}$. For $y = x^{-3}$, we have $\frac{dy}{dx}=-3x^{-3 - 1}=-3x^{-4}$.

Step3: Convert to positive exponents

Using the negative - exponent rule $a^{-n}=\frac{1}{a^{n}}$, we rewrite $-3x^{-4}$ as $-\frac{3}{x^{4}}$.

Answer:

$-\frac{3}{x^{4}}$