Imagine you're faced with a function like this:
Up to this point in our studies, we’ve primarily focused on straightforward single-variable derivatives. This function, however, involves both x and y on one side of the equation!
Whenever you're faced with a function with both x and y on one side of the equation, utilize implicit differentiation to take the derivative!
How to solve with implicit differentiation
A general outline of implicit differentiation problems goes as the following:
- Evaluate the derivative on both sides of the expression
- Find all terms that include the derivative of y (dy/dx)
- Isolate the derivative of y (dy/dx) terms on one side of the equation
Let's go through each of these steps in the problem above to learn how to apply implicit differentiation!
Evaluate the derivative on both sides of the expression
Let's start with the left-side.
Lucky for us, the left-side of the equation is just y.
To take the derivative of y, we're going to apply the following truth:
Whenever taking the derivative of y in implicit differentiation, treat it as x. Then, multiply it by the derivative of y, a.k.a. (dy/dx)!
Let's start by treating this y as x. Simply put, the derivative of x is 1. Therefore, the derivative of y here will be 1 too...
...but wait! We've then gotta multiply this by (dy/dx), since we're taking the derivative of a y term!
That's it for the left-side! Now, let's take the derivative of the right-side.
To take the derivative of the right side...
...we're going to apply the sum rule. In essence, this means we'll take the derivative of each term here individually.