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3.2 Implicit Differentiation

#derivatives

#implicit

โฑ๏ธย ย 1 min read

written by

Jillian Holbrook

jillian holbrook


What is Implicit Differentiation?

๐ŸŽฅWatch: 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.ย  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.ย  Most of the time, an implicit equation will have x and y on the same side.ย  So if x and y are on the same side, how can we differentiate an implicit equation?

How to Differentiate Implicitly ๐Ÿ•ต

First off, letโ€™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 fโ€™(x) notation.ย  Instead, we will use the dy/dx and y' notations.

There are three main steps to successfully differentiate an equation implicitly.

  1. Get the yโ€™s isolated on one side

  2. Factor out yโ€™

  3. Isolate yโ€™

Letโ€™s look at an example to apply these steps.ย ย 

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 ๐Ÿ“

Youโ€™ll be a pro at implicitly differentiating in no time! Try the practice problems below and check them when youโ€™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 ๐Ÿ“

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

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