Cracking The Ap Calculus ab Exam 2018

y^2     +   y   =   3x^5    −   7x

You should  use implicit    differentiation any time    you can’t   write   a

function    explicitly  in  terms   of  the variable    that    we  want    to  take

the derivative  with    respect to.

To take the derivative according to the information in the last paragraph, you get

Notice how each variable is multiplied by its appropriate . Now, remembering that = 1, rewrite the

expression this way: = 15x^4 − 7.

Next, factor out of the left-hand side: (2y + 1) = 15x^4 − 7.

Isolating gives you = .

This is the derivative you’re looking for. Notice how the derivative is defined in terms of y and x. Up

until now, has been strictly in terms of x. This is why the differentiation is “implicit.”

Confused? Let’s do a few examples and you will get the hang of it.

Example 1: Find if y^3 − 4y^2 = x^5 + 3x^4.

Using implicit differentiation, you get

Remember that = 1: (3y^2 − 8y) = 5x^4 + 12x^3.