Implicit Differentiation

In the last section, we learned how to take derivatives of some trigonometric functions namely, the sine, cosine, and tangent. In this section, we are going to learn a different way to differentiate which is not like chain rule, the product rule, or any other rule we have learned about so far. This new way to take the derivative is called implicit differentiation. The word 'implicit' should make you think implied, suggesting. Suppose we had a function that took the following form:

With implicit differentiation, we differentiate the x variable just as we would if we were doing a typical differentiation, but this time we different the y along with it like this:

Now we have , and we can solve for it as if it were any other variable. Solving for we have:

We can check our answer by solving for y in the original equation, and then differentiating normally. Solving for y our equation looks like this which is the equation of a parabola shifted to the left two units. We can take the derivative using the chain rule, and you can see our answers are the same.

Let's try implicit differentiation on one more problem. Suppose we had an equation that looked like this: . First we differentiate both sides of the equation and have something that looks like this: . Solving for we have which is the derivative of y with respect to x. That's all there is to basic implicit differentiation.