# How do you implicitly differentiate -3=(xy)/(x-e^y)?

##### 1 Answer
Dec 23, 2016

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{1}{x}$

#### Explanation:

Let's write the equation in the form:

$- 3 x + 3 {e}^{y} = x y$

and differentiate term to term with respect to $x$:

$- 3 + 3 {e}^{y} \frac{\mathrm{dy}}{\mathrm{dx}} = y$

no solve for the derivative:

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{y + 3}{3 {e}^{y}}$

From the initial equation we can derive:

$3 {e}^{y} = x y + 3 x = x \left(y + 3\right)$

and substituting this in the expression of the derivative we get:

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{y + 3}{3 {e}^{y}} = \frac{y + 3}{x \left(y + 3\right)} = \frac{1}{x}$