# Question #0caf6

Feb 6, 2017

The curve has a slope of $- 1$ at $\left(2 , 0\right)$.

#### Explanation:

The trick is to always remember you are differentiating with respect to $x$ and to write "$\frac{\mathrm{dy}}{\mathrm{dx}}$" whenever you differentiate a different variable.

We differentiate the above equation using the power rule, which states that for a function $f \left(x\right) = {x}^{n}$, the derivative is given by $f ' \left(x\right) = n {x}^{n - 1}$.

$2 \left(\frac{1}{2}\right) x + 2 y \left(\frac{\mathrm{dy}}{\mathrm{dx}}\right) + 2 \left(\frac{\mathrm{dy}}{\mathrm{dx}}\right) - 0 = 0$

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

Solve for $\frac{\mathrm{dy}}{\mathrm{dx}}$

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

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

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

The slope of the curve at a point $x = a$ can be obtained by evaluating $x = a$ within $\frac{\mathrm{dy}}{\mathrm{dx}}$.

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

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

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

Hopefully this helps!