After you factor out , divide both sides by 3y^2 − 8y.
Note: Now that you understand that the derivative of an x term with respect to x will always be multiplied
by , and that = 1, we won’t write anymore. You should understand that the term is implied.
Example 2: Find if sin y^2 − cos x^2 = cos y^2 + sin x^2.
Use implicit differentiation.
cos y^2 + sin x^2 (2x) = −sin y^2 + cos x^2 (2x)Then simplify.
2 y cos y^2 + 2 x sin x^2 = −2 y sin y^2 + 2 x cos x^2Next, put all of the terms containing on the left and all of the other terms on the right.
2 y cos y^2 + 2 y sin y^2 = −2 x sin x^2 + 2x cos x^2Next, factor out .
(2 y cos y^2 + 2 y sin y^2 ) = −2x sin x^2 + 2x cos x^2And isolate .
This can be simplified further to the following: