# Question #518f7

##### 1 Answer
Jan 10, 2018

$\frac{{d}^{2} y}{\mathrm{dx}} ^ 2 = \frac{{y}^{2} - {x}^{2}}{y} ^ 3$

#### Explanation:

To find $\frac{{d}^{2} y}{\mathrm{dx}} ^ 2$, we must first find $\frac{\mathrm{dy}}{\mathrm{dx}}$ which we can do using implicit differentiation.

$\frac{d}{\mathrm{dx}} \left[{x}^{2} - {y}^{2}\right] = \frac{d}{\mathrm{dx}} \left[5\right]$
$\frac{d}{\mathrm{dx}} {x}^{2} - \frac{d}{\mathrm{dx}} {y}^{2} = \frac{d}{\mathrm{dx}} 5$ (Sum Rule)
$2 x - 2 y \frac{\mathrm{dy}}{\mathrm{dx}} = 0$
$2 x = 2 y \frac{\mathrm{dy}}{\mathrm{dx}}$
$\frac{2 x}{2 y} = \frac{\mathrm{dy}}{\mathrm{dx}}$
$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{x}{y}$

Now, we can find the second derivative.

$\frac{d}{\mathrm{dx}} \left[\frac{\mathrm{dy}}{\mathrm{dx}}\right] = \frac{d}{\mathrm{dx}} \left[\frac{x}{y}\right]$
$\frac{{d}^{2} y}{\mathrm{dx}} ^ 2 = \frac{y \frac{d}{\mathrm{dx}} x - x \frac{d}{\mathrm{dx}} y}{y} ^ 2$ (Quotient Rule)
$\frac{{d}^{2} y}{\mathrm{dx}} ^ 2 = \frac{y - x \frac{\mathrm{dy}}{\mathrm{dx}}}{y} ^ 2$
$\frac{{d}^{2} y}{\mathrm{dx}} ^ 2 = \frac{y - x \left(\frac{x}{y}\right)}{y} ^ 2$
$\frac{{d}^{2} y}{\mathrm{dx}} ^ 2 = \frac{y - {x}^{2} / y}{y} ^ 2$
$\frac{{d}^{2} y}{\mathrm{dx}} ^ 2 = \frac{{y}^{2} - {x}^{2}}{y} ^ 3$