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What is implicit differentiation and how do I do it?

1 min readโ€ขdecember 17, 2021


AP Calculus AB/BCย โ™พ๏ธ

279ย resources
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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 calledย explicitย differentiation.
Implicit differentiation, on the other hand, is differentiating a variableย in terms of another variable. We're not just taking the derivative ofย xย orย 8x+6ย anymore, we're taking the derivative of whole equations likeย y = 8x+6ย to findย dy/dx.
Need a quick review on taking and finding derivatives, make sure you watch this ๐ŸŽฅ 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:ย 
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?ย Just apply the chain rule!
๐ŸŒŸ 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 forย 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 forย 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 ๐ŸŽฅ implicit differentiation and derivatives.

Browse Study Guides By Unit
๐Ÿ‘‘Unit 1 โ€“ Limits & Continuity
๐Ÿค“Unit 2 โ€“ Fundamentals of Differentiation
๐Ÿค™๐ŸฝUnit 3 โ€“ Composite, Implicit, & Inverse Functions
๐Ÿ‘€Unit 4 โ€“ Contextual Applications of Differentiation
โœจUnit 5 โ€“ Analytical Applications of Differentiation
๐Ÿ”ฅUnit 6 โ€“ Integration & Accumulation of Change
๐Ÿ’ŽUnit 7 โ€“ Differential Equations
๐ŸถUnit 8 โ€“ Applications of Integration
๐Ÿฆ–Unit 9 โ€“ Parametric Equations, Polar Coordinates, & Vector-Valued Functions (BC Only)
โ™พUnit 10 โ€“ Infinite Sequences & Series (BC Only)
๐ŸงMultiple Choice Questions (MCQ)
โœ๏ธFree Response Questions (FRQ)
๐Ÿ“†Big Reviews: Finals & Exam Prep

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