# Question #36537

Oct 30, 2016

$\frac{{d}^{2} y}{\mathrm{dx}} ^ 2 = \frac{3 {x}^{2} \left({y}^{4} - {x}^{4}\right)}{y} ^ 7 = \frac{- 3 {x}^{2}}{y} ^ 7$

#### Explanation:

${x}^{4} - {y}^{4} = 1$

Differentiate both sides of the equation with respect to $x$

$4 {x}^{3} - 4 {y}^{3} \frac{\mathrm{dy}}{\mathrm{dx}} = 0$

So $\frac{\mathrm{dy}}{\mathrm{dx}} = {x}^{3} / {y}^{3}$

Now differentiate again w.r.t. $x$.

$\frac{{d}^{2} y}{\mathrm{dx}} ^ 2 = \frac{3 {x}^{2} {y}^{3} - {x}^{3} 3 {y}^{2} \frac{\mathrm{dy}}{\mathrm{dx}}}{y} ^ 6$

Get rid of the common factor ${y}^{2}$

$= \frac{3 {x}^{2} y - 3 {x}^{3} \frac{\mathrm{dy}}{\mathrm{dx}}}{y} ^ 4$

Replace $\frac{\mathrm{dy}}{\mathrm{dx}} = {x}^{3} / {y}^{3}$

$= \frac{3 {x}^{2} y - 3 {x}^{3} {x}^{3} / {y}^{3}}{y} ^ 4$

Clear the fraction in the numerator by multiplying by ${y}^{3} / {y}^{3}$

$= \frac{3 {x}^{2} {y}^{4} - 3 {x}^{6}}{y} ^ 7$

Factor the numerator.

$= \frac{3 {x}^{2} \left({y}^{4} - {x}^{4}\right)}{y} ^ 7$

Recognize the opposite of the initial expression.

${x}^{4} - {y}^{4} = 1 \iff {y}^{4} - {x}^{4} = - 1$