---
title: "Implicit Differentiation and Derivatives of Inverse Functions"
description: "Differentiate a relation that is not solved for y, and use the same move to get the derivative of an inverse function from the derivative of the original."
canonical: https://lightmysky.com/learn/mathematics/implicit-differentiation-and-derivatives-of-inverse-functions-mt_P4ABBGsKFT
source: https://lightmysky.com/learn/mathematics/implicit-differentiation-and-derivatives-of-inverse-functions-mt_P4ABBGsKFT.md
retrieved: 2026-09-12
---

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# Implicit Differentiation and Derivatives of Inverse Functions

Differentiate a relation that is not solved for y, and use the same move to get the derivative of an inverse function from the derivative of the original.

Subject: Mathematics · Area: Calculus & Analysis · Ages 18 to 19
Page: https://lightmysky.com/learn/mathematics/implicit-differentiation-and-derivatives-of-inverse-functions-mt_P4ABBGsKFT

## Ready when they can

- Find dy/dx for x² + y² = 25 and read the gradient at a point on the circle
- Derive the derivative of arcsin x from sin y = x
- Explain why dy/dx appears as a factor whenever y is differentiated

## Lesson: Slopes without solving for y

Some curves never solve neatly for y, like the circle x squared + y squared = 25. Differentiate both sides with respect to x and treat y as a function of x, so y squared gives 2y times dy/dx. Collecting gives dy/dx = -x/y, so at (3, 4) the gradient is -0.75 and the tangent is y = -0.75x + 6.25.

Inverse derivatives fall out the same way. Write y = arcsin x as sin y = x, differentiate to cos y times dy/dx = 1, and trade cos y for sqrt(1 - x squared) to get 1/sqrt(1 - x squared). Arccos takes the negative twin, arctan x gives 1/(1 + x squared), and arctan(3x) adds a factor 3. In general (f inverse)' = 1 / f'(f inverse(x)), so f(2) = 5 with f'(2) = 3 gives inverse gradient 1/3 at 5.

**Example.** For xy = 12 at (3, 4), the product rule gives y + x times dy/dx = 0, so dy/dx = -y/x = -4/3. For x squared + xy + y squared = 37 at (3, 4), collecting gives dy/dx = -(2x + y)/(x + 2y) = -10/11, about -0.91.

**Recap.** Differentiate each y term with its dy/dx attached, gather them, divide, then put in the point.

## Practice

20 questions on this page, each with its working shown.

## Needs first

- [Derivatives of Exponential, Logarithmic and Trigonometric Functions](https://lightmysky.com/learn/mathematics/derivatives-of-exponential-logarithmic-and-trigonometric-functions-mt_695WPZKADT)
- [Inverse Functions](https://lightmysky.com/learn/mathematics/inverse-functions-mt_7k7q5ZjMb4)

## Opens up

- [Related Rates](https://lightmysky.com/learn/mathematics/related-rates-mt_3XTIPt8Vmf)