Derivative of 1/x^3: Answer, Proof, and Mistakes

The proof: rewrite 1/x^3 as a power

Move the cube to the numerator as a negative exponent: $\frac{1}{x^3} = x^{-3}$. Now the power rule $\frac{d}{dx}x^n = n\,x^{n-1}$ applies directly with $n = -3$.

The exponent $-3$ comes down as the coefficient, and the new exponent is $-3 - 1 = -4$. That leading $-3$ is where the minus sign in the answer comes from.

The slope is negative wherever the function is defined

For every $x \neq 0$, $x^4 > 0$, so $-\frac{3}{x^4}$ is negative. The slope is never zero and never positive, so $\frac{1}{x^3}$ has no critical points and decreases on each of its two branches.

As $|x|$ grows the slope flattens toward $0$; as $x$ approaches $0$ the slope steepens sharply, matching the vertical asymptote at $x = 0$ where the function is undefined.

Common mistakes

• Dropping the minus sign and writing $\frac{3}{x^4}$. The derivative is negative everywhere, so a positive answer is wrong on sight.
• Getting the new exponent wrong: from $x^{-3}$ you subtract $1$ to reach $x^{-4}$, not $x^{-2}$. Subtracting from a negative exponent is the common slip.
• Leaving the coefficient as $1$ or forgetting to bring the exponent down, giving $-x^{-4}$ instead of $-3x^{-4}$.
• Treating $\frac{1}{x^3}$ as a quotient and misapplying the quotient rule; the power-rule route is faster and less error-prone.