Question

What is the implicit derivative of `3=e^y-e^(2y)`?

Answer 1

Hey there!

To compute any implicit derivative, you always want to differentiate like you normally would with respect to x, but when taking the derivative of something with a "y" in it, you must multiply by y' or dy/dx. After that, isolate for dy/dx, then you're done! If you want to take the derivative with respect to it's the same idea!

Explanation:

In your example you have `3 = e^y - e^(2y)`:

First, take the derivative of everything with respect to x, but remember, when taking the derivative of a term with "y" in it, you must multiply by dy/dx:

`0 = e^y*dy/dx - 2e^(2y)*dy/dx`

Factor out dy/dx:

`0 = dy/dx(e^y - e^(2y))`

Divide by `e^y - e^(2y)`on both sides:

`0 = dy/dx`

That's the derivative (with respect to x)! Hopefully that helps! All in all, that's the process you should follow do calculate implicit derivatives!