# What is the implicit derivative of 1= xe^(4y?

Dec 2, 2015

$- \frac{1}{4 x}$

#### Explanation:

To find an Implicit Derivative, differentiate like normal except remember that when you differentiate a $y$ there will be an extra $\frac{\mathrm{dy}}{\mathrm{dx}}$ term, thanks to the Chain Rule.

d/dx[1=xe^(4y)]

$0 = {e}^{4 y} \frac{d}{\mathrm{dx}} \left[x\right] + x \frac{d}{\mathrm{dx}} \left[{e}^{4 y}\right]$

Find each derivative separately:

$\frac{d}{\mathrm{dx}} \left[x\right] = 1$

$\frac{d}{\mathrm{dx}} \left[{e}^{4 y}\right] = {e}^{4 y} \frac{d}{\mathrm{dx}} \left[4 y\right]$

$\frac{d}{\mathrm{dx}} \left[4 y\right] = 4 \frac{\mathrm{dy}}{\mathrm{dx}}$

So, $\frac{d}{\mathrm{dx}} \left[{e}^{4 y}\right] = 4 {e}^{4 y} \frac{\mathrm{dy}}{\mathrm{dx}}$

Plug them back in:

$0 = {e}^{4 y} + 4 x {e}^{4 y} \frac{\mathrm{dy}}{\mathrm{dx}}$

$- {e}^{4 y} = 4 x {e}^{4 y} \frac{\mathrm{dy}}{\mathrm{dx}}$

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{- {e}^{4 y}}{4 x {e}^{4 y}} = - \frac{1}{4 x}$