Feb 6, 2018

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{2}{x} ^ 2$

#### Explanation:

You may be tempted to use implicit differentiation here, but since you have a relatively simple equation, it's much easier to solve for $y$ in terms of $x$, and then just use normal differentiation. So:

$2 + x y = x$

$\implies y = \frac{x - 2}{x} = 1 - \frac{2}{x}$

Now we just use a simple power rule:

$\implies \frac{\mathrm{dy}}{\mathrm{dx}} = - \left(- 2 {x}^{-} 2\right)$

$= \frac{2}{x} ^ 2$

There you are! Note that you could have used implicit differentiation to solve this, but by doing this we have a derivative that's in terms of just $x$, which is slightly more convenient. However, regardless of the method you use, your answer should be the same.

Hope that helped :)