Implicit Differentiation
Differentiate both sides when y is tangled with x.
About this topic
When y is tied to x in an equation that is hard to solve for y, such as x² + y² = 25, differentiate both sides with respect to x, treating y as a function of x and applying the chain rule, so d/dx(y²) = 2y·dy/dx. Then solve for dy/dx.
For the circle you get 2x + 2y·dy/dx = 0, so dy/dx = −x/y. At the point (3, 4) the slope of the tangent is −3/4, which agrees with the geometry that the tangent is perpendicular to the radius. The method handles curves that fail the vertical line test, such as circles and ellipses, and equations mixing x and y like x³ + y³ = 6xy.
The key steps are: differentiate every term, attach dy/dx whenever you differentiate a y-term, use the product rule for terms such as xy, collect all dy/dx terms on one side, and factor it out.
Implicit differentiation is essential for related rates and for finding derivatives of inverse functions such as ln x and arcsin x. Common mistakes include forgetting the dy/dx factor and forgetting the product rule on xy, which gives y + x·dy/dx.
Key ideas
- Treat y as a function of x
- Multiply by dy/dx after differentiating y
- Solve for dy/dx
Quick check
Use cases: where Implicit Differentiation helps
- Homework: review the key idea behind Implicit Differentiation before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Implicit Differentiation without notes.
- Visual learning: play with the interactive lab to see how Implicit Differentiation changes when you adjust its inputs.
- Teaching a friend: explain Implicit Differentiation in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Implicit Differentiation?
Differentiate both sides when y is tangled with x.
What will I learn in Implicit Differentiation?
Treat y as a function of x. Multiply by dy/dx after differentiating y. Solve for dy/dx.
Is Implicit Differentiation part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Implicit Differentiation lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Implicit Differentiation?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.