Differential Equations For Dummies
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Implicit differentiation problems are chain rule problems in disguise. Here's why: You know that the derivative of sin x is cos x, and that according to the chain rule, the derivative of sin (x3) is

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You could finish that problem by doing the derivative of x3, but there is a reason for you to leave the problem unfinished here.

To do implicit differentiation, all you do (sort of) is every time you see a "y" in a problem, you treat it like the x3 is treated here. Thus, because the derivative of sin (x3) is

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the derivative of sin y is

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Then, after doing the differentiation,

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By the way, "y" is used in the preceding explanation, but that's not the whole story. Consider that

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is the same as

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It's the variable on the top that you apply implicit differentiation to. This is typically y, but it could be any other variable. And it's the variable on the bottom that you treat the ordinary way. This is typically x, but it could also be any other variable.

Okay, time for a few practice questions.

Practice questions

  1. if y3x2 = x + y, find

    image6.png
  2. For x2y = y3x + 5y + x, find

    image7.png

Answers and explanations

  1. By implicit differentiation,

    image8.png

    Start by taking the derivative of all four terms, using the chain rule (sort of) for all terms containing a y.

    image9.png

    Then move all terms containing

    image10.png

    to the left, move all other terms to the right, and factor out

    image11.png

    Divide and voilà!

    image12.png
  2. By implicit differentiation,

    image13.png

    This time you have two products to deal with, so use the product rule for the two products and the regular rules for the other two terms.

    image14.png

About This Article

This article is from the book:

About the book author:

Steven Holzner was an award-winning author of more than 130 books, of which more than 2 million copies have been sold. His books have been translated into 23 languages. He served on the Physics faculty at Cornell University for more than a decade, teaching both Physics 101 and Physics 102. Holzner received his doctorate in physics from Cornell and performed his undergraduate work at Massachusetts Institute of Technology, where he also served as a faculty member.

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