# How do you use implicit differentiation to find the slope of the curve given x^2+xy+y^2=7 at (3,2)?

Oct 26, 2016

See below.

#### Explanation:

This point is not actually on the graph. Let's pick the point $\left(3 , - 1\right)$.

We start by differentiating, using the product rule, the power rule and implicit differentiation.

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

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

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

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{- y - 2 x}{x + 2 y}$

The slope of the curve is given by evaluating the point within the derivative.

So, letting the slope be $m$, we have:

$m = \frac{- \left(- 1\right) - 2 \left(3\right)}{3 + 2 \left(- 1\right)}$

$m = \frac{1 - 6}{3 - 2}$

$m = - 5$

Hopefully this helps!