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3.2 Implicit Differentiation

1 min readβ€’june 7, 2020

Jillian Holbrook

Jillian Holbrook


AP Calculus AB/BC ♾️

279Β resources
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What is Implicit Differentiation?

πŸŽ₯Watch: AP Calculus AB/BC - Implicit Derivatives
Back in pre-calculus, you likely learned or talked about how there are two different types of equations: explicit equations and implicit equations.
An explicit equation is written such that y is isolated on one side.Β  For example, y=2x+3 is an explicit equation. We know that for each value of x, there will only be one value of y.
Implicit equations are not as straightforward.Β  Most of the time, an implicit equation will have x and y on the same side.Β  So if x and y are on the same side, how can we differentiate an implicit equation?

How to Differentiate Implicitly πŸ•΅

First off, let’s review the notations we can use for derivatives:
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(450).png?alt=media&token=467543be-5b3a-4cd7-ac3f-3d291e0bf461
Implicit Differentiation does not use the f’(x) notation.Β  Instead, we will use the dy/dx and y' notations.
There are three main steps to successfully differentiate an equation implicitly.
  1. Get the y’s isolated on one side
  2. Factor out y’
  3. Isolate y’
Let’s look at an example to apply these steps.Β Β 
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(452).png?alt=media&token=4390de53-e542-4ccd-92fb-2dcbd4f20d5c

Implicit Differentiation Practice πŸ“

You’ll be a pro at implicitly differentiating in no time! Try the practice problems below and check them when you’re finished in the corresponding section at the end of this guide.
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(455).png?alt=media&token=28754308-41fd-40e9-aa52-1dc3a537c4bb

Answers πŸ“

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreen%20Shot%202020-05-19%20at%202.24-i77L644lkzG7.png?alt=media&token=674bb9fa-78eb-49fc-a9f7-b1dfdc0208f3
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreen%20Shot%202020-05-19%20at%202.24-CclBnRizK2Kl.png?alt=media&token=03a8657a-72bf-47c2-9837-85ec66387fc9
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)
🧐Multiple Choice Questions (MCQ)
✍️Free Response Questions (FRQ)
πŸ“†Big Reviews: Finals & Exam Prep

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