Derivative of an inverse function

When $x$ is written in terms of $y$ (say $x=y^3$), differentiate with respect to $y$ first, $\frac{dx}{dy}=3y^2$, then take the reciprocal.

This agrees with differentiating $y=x^{1/3}$ directly, $\frac{1}{3}x^{-2/3}$, rewritten in terms of $y$. For $x=e^y$, $\frac{dx}{dy}=e^y$, so $\frac{dy}{dx}=\frac{1}{e^y}=\frac{1}{x}$, which is where $(\ln x)'=\frac{1}{x}$ comes from.

Whether to leave the answer in $y$ or convert to $x$ depends on the problem.

Examples

1. 01What is dy/dx?$\displaystyle x=y^3$AnswerHide

   Answer$\displaystyle \frac{1}{3y^2}$
2. 02What is dy/dx?$\displaystyle x=2y$AnswerHide

   Answer$\displaystyle \frac{1}{2}$
3. 03What is dy/dx?$\displaystyle x=y^2$AnswerHide

   Answer$\displaystyle \frac{1}{2y}$
4. 04What is dy/dx?$\displaystyle x=y^5$AnswerHide

   Answer$\displaystyle \frac{1}{5y^4}$
5. 05What is dy/dx?$\displaystyle x=e^y$AnswerHide

   Answer$\displaystyle \frac{1}{e^y}$
6. 06What is dy/dx?$\displaystyle x=\sin y$AnswerHide

   Answer$\displaystyle \frac{1}{\cos y}$

FAQ

Q1Why is it just the reciprocal?Show answerHide

A

Because $\Delta y/\Delta x$ and $\Delta x/\Delta y$ are reciprocals. Where $dx/dy\neq 0$ this survives the limit, giving $\frac{dy}{dx}=\frac{1}{dx/dy}$.

Practice this pattern in the app, 5 problems a day.

Published 2026-09-04 · Updated 2026-09-05