# If y^2 - 2xy = 21, what is the slope of the tangent at (2, -3)?

Jan 21, 2017

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{3}{5}$ at $\left(2 , - 3\right)$.

#### Explanation:

Differentiate using implicit differentiation and the product rule. Remember that we are differentiating with respect to $x$.

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

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

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

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

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

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

Simply evaluate now:

$\frac{\mathrm{dy}}{\mathrm{dx}} = - \frac{3}{- 3 - 2}$

$\frac{\mathrm{dy}}{\mathrm{dx}} = - \frac{3}{-} 5$

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{3}{5}$

Hopefully this helps!