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

##### 1 Answer

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 $3 = {e}^{y} - {e}^{2 y}$:

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:

$0 = {e}^{y} \cdot \frac{\mathrm{dy}}{\mathrm{dx}} - 2 {e}^{2 y} \cdot \frac{\mathrm{dy}}{\mathrm{dx}}$

Factor out dy/dx:

$0 = \frac{\mathrm{dy}}{\mathrm{dx}} \left({e}^{y} - {e}^{2 y}\right)$

Divide by ${e}^{y} - {e}^{2 y}$on both sides:

$0 = \frac{\mathrm{dy}}{\mathrm{dx}}$

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!
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