πŸ“š

 > 

♾️ 

 > 

✏️

What is implicit differentiation and how do I do it?

1 min readβ€’july 11, 2024


AP Calculus AB/BC β™ΎοΈ

279 resources
See Units

Explicit vs. Implicit Differentiation

By now, you've probably covered basic differentiation of a function y in terms of a single variable x. This is called explicit differentiation.
Implicit differentiation, on the other hand, is differentiating a variable in terms of another variable. We're not just taking the derivative of x or 8x+6 anymore, we're taking the derivative of whole equations like y = 8x+6 to find dy/dx.
Need a quick review on taking and finding derivatives, make sure you watch this πŸŽ₯ video introducing and explaining derivatives for a refresher.

Explicit Differentiation

  • Single variable function/relation
Example:
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-rQZl3VlhZ7Dz.gif?alt=media&token=42a46c0e-abd1-4058-823f-fa7cca3a5175

Implicit Differentiation

  • More than one variable in the function/relation
Example: 
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-YaaPr7SGun2u.gif?alt=media&token=82697ac6-a706-440d-bb57-18d80a801595
So how do you do implicit differentiation? Just apply the chain rule!
🌟 Example:
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-dNXB1MAmqAgi.gif?alt=media&token=0697d204-69c1-4ebe-b034-1063bef45278
We can even simplify further to solve for dy/dx!
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-mewPe7iPAiti.gif?alt=media&token=85c6c348-6625-40bb-ad9e-558eb835ea51
...And there you have it!

Implicit differentiation is useful in solving differential equations, where you'll need to solve for dy/dx. Some applications include optimization, e.g. finding the rate of change of volume with respect to the rate of change of time.
For more examples and help watch this video about πŸŽ₯ implicit differentiation and derivatives.

Browse Study Guides By Unit
πŸ‘‘Unit 1 – Limits & Continuity
πŸ€“Unit 2 – Fundamentals of Differentiation
πŸ€™πŸ½Unit 3 – Composite, Implicit, & Inverse Functions
πŸ‘€Unit 4 – Contextual Applications of Differentiation
✨Unit 5 – Analytical Applications of Differentiation
πŸ”₯Unit 6 – Integration & Accumulation of Change
πŸ’ŽUnit 7 – Differential Equations
🐢Unit 8 – Applications of Integration
πŸ¦–Unit 9 – Parametric Equations, Polar Coordinates, & Vector-Valued Functions (BC Only)
β™ΎUnit 10 – Infinite Sequences & Series (BC Only)
πŸ“šStudy Tools
πŸ€”Exam Skills

Fiveable
Fiveable
Home
Stay Connected

Β© 2025 Fiveable Inc. All rights reserved.


Β© 2025 Fiveable Inc. All rights reserved.