Example 3: Find if 3x^2 + 5xy^2 − 4y^3 = 8.
Implicit differentiation should result in the following:
Did you notice the use of the Product Rule to find the derivative
of 5xy 2 ? The AP Exam loves to make you do this. All of the
same differentiation rules that you’ve learned up until now still
apply. We’re just adding another technique.You can simplify this to
6 x + 10xy + 5 y^2 − 12 y^2 = 0Next, put all of the terms containing on the left and all of the other terms on the right.
10 xy − 12 y^2 = −6x − 5 y^2Next, factor out .
(10xy − 12 y^2 ) = −6x −5 y^2Then, isolate .
Example 4: Find the derivative of 3x^2 − 4y^2 + y = 9 at (2, 1).