Question

Find d2y/dx2 in terms of x and y. y^4 = x^7

Answer 1

$y^4=x^7$

Differentiating once gives

$4y^3\dfrac{\mathrm dy}{\mathrm dx}=7x^6$

Differentiating again gives

$12y^2\left(\dfrac{\mathrm dy}{\mathrm dx}\right)^2+4y^3\dfrac{\mathrm d^2y}{\mathrm dx^2}=42x^5$

From the first result, you get

$4y^3\dfrac{\mathrm dy}{\mathrm dx}=7x^6\implies\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{7x^6}{4y^3}$

and plugging this into the second gives

$12y^2\left(\dfrac{7x^6}{4y^3}\right)^2+4y^3\dfrac{\mathrm d^2y}{\mathrm dx^2}=42x^5$
$12y^2\dfrac{49x^{12}}{16y^6}+4y^3\dfrac{\mathrm d^2y}{\mathrm dx^2}=42x^5$
$\dfrac{147x^{12}}{4y^4}+4y^3\dfrac{\mathrm d^2y}{\mathrm dx^2}=42x^5$

and solving for $\dfrac{\mathrm d^2y}{\mathrm dx^2}$ gives

$\dfrac{\mathrm d^2y}{\mathrm dx^2}=\dfrac{42x^5-\frac{147x^{12}}{4y^4}}{4y^3}$
$\dfrac{\mathrm d^2y}{\mathrm dx^2}=\dfrac{168x^5y^4-147x^{12}}{16y^7}$