Implicit Differentiation — Practice Questions

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

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

the derivative of sin y is

Then, after doing the differentiation,

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

is the same as

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 y³ – x² = x + y, find

   

2. For x²y = y³x + 5y + x, find

   

Answers and explanations

1. By implicit differentiation,

   

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

   

   Then move all terms containing

   

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

   

   Divide and voilà!

   

2. By implicit differentiation,

   

   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.