# How do you differentiate e^(x / y) = 8x - y?

Sep 26, 2015

Make use of the method of implicit differentiation to obtain
$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{8 + y {e}^{\frac{x}{y}}}{1 - x {e}^{\frac{x}{y}}}$

#### Explanation:

Since we cannot obtain y as an explicit function of x, we will use the method of implicit differentiation to find the derivative of y with respect to x.
In so doing, we must bear in mind that y is a function of x.

We rearrange the expression and then take the derivative on both sides using normal rules of differentiation to obtain

$y = 8 x - {e}^{\frac{x}{y}}$

$\therefore \frac{\mathrm{dy}}{\mathrm{dx}} = 8 - {e}^{\frac{x}{y}} \cdot \left(x \cdot \frac{\mathrm{dy}}{\mathrm{dx}} - y \cdot 1\right)$

Note we used the quotient rule in the last term since y is a function of x so x/y represents a quotient of 2 functions in x.

Rearranging we get

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{8 + y {e}^{\frac{x}{y}}}{1 - x {e}^{\frac{x}{y}}}$