How do you define the rate of change when your function has variables that cannot be separated? Learn how implicit differentiation can be used to find dy/dx even when you don’t have y=f(x)!

## Implicit Function

I have an Uncle Joe who’s a farmer.

He really likes math, so he told me about his plot of land. He said his land extends *x* meters to the east and *y* meters to the north. He said, ‘You know, the area of that land I have is (*x*)(*y*).’ That’s because his land is just a rectangle, and the border, or perimeter, around his land is just 2*x* + 2*y*; we’ve got *x* + *y* + *x* + *y*. Now Uncle Joe told me that his land always satisfies one condition. That is that the area of his land is always equal to half of the perimeter.

In other words, (*x*)(*y*) = 1/2(2*x* + 2*y*) = *x* + *y*. That’s great, Uncle Joe! You love math – what do you need me for?Well, Uncle Joe always wants this equation, (*x*)(*y*) = *x* + *y*, to be true. He’s in the land business. He wants to know if he buys more land to the east – so if he changes *x* – how much does he have to change *y* to keep this equation true? He wants to know *dy/dx* – how much *y* should change while *x* is changing. Oh, so Uncle Joe wants me to calculate a derivative. I can do this.

## Derivatives of an Implicit Function

divide by zero, and we can’t do that.
This is how we find ## Lesson Summary
- Differentiate both sides.
- Collect
*y`*terms to one side of the equation. - Factor
*y`*out of those terms. - Solve for
*y`*.
We end up with |