How left-hand side of my equation becomes y

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

Diagram of the land owned by Uncle Joe
Uncle Joes Land

I have an Uncle Joe who’s a farmer.

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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 2x + 2y; 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(2x + 2y) = 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

Finding y prime in the Uncle Joe example
In the final example, use the product rule on the first term, ye^x
divide by zero, and we can’t do that.

Solving for y prime in the final example
Implicit Function Derivative Example 2 Y Prime

This is how we find y`, or dy/dx, for another case where we have an implicit equation, y = ye^x + x.

Lesson Summary

Implicit differentiation is what you use when you have x and y on both sides of an equation and you’re looking for dy/dx. We did this in the case of Farmer Joe’s land when he gave us the equation (x)(y) = x + y.To do implicit differentiation, we:

  1. Differentiate both sides.
  2. Collect y` terms to one side of the equation.
  3. Factor y` out of those terms.
  4. Solve for y`.

We end up with y` as some function of x and y.


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