Implicit Differentiation — Practice Questions - dummies

Implicit Differentiation — Practice Questions

By Mark Ryan

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

image1.png

the derivative of sin y is

image2.png

Then, after doing the differentiation,

image3.png

By the way, “y” is used in the preceding explanation, but that’s not the whole story. Consider that

image4.png

is the same as

image5.png

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