# How do you use implicit differentiation to find (d^2y)/(dx^2) given 5=4x^2+5y^2?

Sep 4, 2016

See below.

#### Explanation:

$4 {x}^{2} + 5 {y}^{2} = 5$

Differentiate both sides of the equation.

$\frac{d}{\mathrm{dx}} \left(4 {x}^{2} + 5 {y}^{2}\right) = \frac{d}{\mathrm{dx}} \left(5\right)$

$8 x + 10 y \frac{\mathrm{dy}}{\mathrm{dx}} = 0$

I prefer to solve for $\frac{\mathrm{dy}}{\mathrm{dx}}$ before differentiating again.

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{- 4 x}{5 y} = - \frac{4}{5} \left(\frac{x}{y}\right)$

Now differentiate again

$\frac{d}{\mathrm{dx}} \left(\frac{\mathrm{dy}}{\mathrm{dx}}\right) = - \frac{4}{5} \frac{d}{\mathrm{dx}} \left(\frac{x}{y}\right)$

$\frac{{d}^{2} y}{\mathrm{dx}} ^ 2 = - \frac{4}{5} \left(\frac{1 y - x \frac{\mathrm{dy}}{\mathrm{dx}}}{y} ^ 2\right)$

Now replace $\frac{\mathrm{dy}}{\mathrm{dx}}$

$\frac{{d}^{2} y}{\mathrm{dx}} ^ 2 = - \frac{4}{5} \left(\frac{1 y - x \frac{- 4 x}{5 y}}{y} ^ 2\right)$

Simplify

$= - \frac{4}{5} \left(\frac{y + \frac{4 x}{5 y}}{y} ^ 2\right)$

$= - \frac{4}{5} \left(\frac{\left(\left(y + \frac{4 x}{5 y}\right) \cdot 5 y\right)}{{y}^{2} \cdot 5 y}\right)$

$= - \frac{4}{5} \left(\frac{5 {y}^{2} + 4 {x}^{2}}{5 {y}^{3}}\right)$

And we're kind of done except that in the original statement of the question we're told that that numerator is $5$.

$4 {x}^{2} + 5 {y}^{2} = 5$, so we can simplify further:

$\frac{{d}^{2} y}{\mathrm{dx}} ^ 2 = - \frac{4}{5 {y}^{3}}$