# How do you find the second derivative implicitly of x^2 + xy = 5?

Aug 31, 2015

$\frac{{d}^{2} y}{\mathrm{dx}} ^ 2 = \frac{10}{x} ^ 3$

#### Explanation:

Implicit differentiation first time is, $2 x + x \frac{\mathrm{dy}}{\mathrm{dx}} + y = 0$

Note that this gives $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{- y - 2 x}{x}$

Implicit differentiation second time gives,

$2 + x \frac{{d}^{2} y}{\mathrm{dx}} ^ 2 + \frac{\mathrm{dy}}{\mathrm{dx}} + \frac{\mathrm{dy}}{\mathrm{dx}}$=0. Hence,

$x \frac{{d}^{2} y}{\mathrm{dx}} ^ 2 = - 2 - 2 \left(\frac{\mathrm{dy}}{\mathrm{dx}}\right) = - 2 - \frac{2}{x} \left(- y - 2 x\right) = 2 + 2 \frac{y}{x} = 2 + 2 \left(\frac{5}{x} ^ 2 - 1\right) = \frac{10}{x} ^ 2$

$\frac{{d}^{2} y}{\mathrm{dx}} ^ 2 = \frac{10}{x} ^ 3$