# Implicit differentiation

When y is tangled up with x in an equation, differentiate both sides with respect to x and treat y as a function of x, using the chain rule.

Maths (university) · Part 03: Differentiation · Card 9 of 10 · Last updated 2026-10-19

## Analogy

Adjusting a seesaw: change one side and the other must shift to stay balanced.

## Example

**Circle x^2 + y^2 = 25**

Differentiate: 2x + 2y (dy/dx) = 0.
So dy/dx = -x / y.
At (3, 4): slope = -3/4.
Check: 9 + 16 = 25.

## In practice

Gives slopes of circles and other curves that are not graphs of y = f(x).

## Practice

### Flashcards

- **What is the slope of x^2 + y^2 = 25 at (3, 4)?** -3/4, from dy/dx = -x/y.

## Questions people ask

### Why does y get a dy/dx when I differentiate implicitly?

Because y depends on x, so it is a function inside the equation. Differentiating y squared gives 2y times dy/dx, by the chain rule. Terms with only x differentiate as normal. Then collect the dy/dx terms and solve for it.

### Can the slope formula contain both x and y?

Yes. For the circle x squared plus y squared equals 25, the slope is minus x over y, which depends on both. You need a point on the curve, such as (3, 4), to get a number: here the slope is -3/4.

## Related concepts

- [Derivative](https://studyvisually.com/learn/science/maths-university/maths-university-part-3/maths-university-part-3-concept-1)
- [Tangent line](https://studyvisually.com/learn/science/maths-university/maths-university-part-3/maths-university-part-3-concept-2)
- [Rate of change](https://studyvisually.com/learn/science/maths-university/maths-university-part-3/maths-university-part-3-concept-3)
- [Power rule](https://studyvisually.com/learn/science/maths-university/maths-university-part-3/maths-university-part-3-concept-4)

Web page: https://studyvisually.com/learn/science/maths-university/maths-university-part-3/maths-university-part-3-concept-9
Study this card in the app: https://studyvisually.com/app/learn?category=maths-university&card=maths-university-part-3-concept-9