[tex]y^4=x^7[/tex]
Differentiating once gives
[tex]4y^3\dfrac{\mathrm dy}{\mathrm dx}=7x^6[/tex]
Differentiating again gives
[tex]12y^2\left(\dfrac{\mathrm dy}{\mathrm dx}\right)^2+4y^3\dfrac{\mathrm d^2y}{\mathrm dx^2}=42x^5[/tex]
From the first result, you get
[tex]4y^3\dfrac{\mathrm dy}{\mathrm dx}=7x^6\implies\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{7x^6}{4y^3}[/tex]
and plugging this into the second gives
[tex]12y^2\left(\dfrac{7x^6}{4y^3}\right)^2+4y^3\dfrac{\mathrm d^2y}{\mathrm dx^2}=42x^5[/tex]
[tex]12y^2\dfrac{49x^{12}}{16y^6}+4y^3\dfrac{\mathrm d^2y}{\mathrm dx^2}=42x^5[/tex]
[tex]\dfrac{147x^{12}}{4y^4}+4y^3\dfrac{\mathrm d^2y}{\mathrm dx^2}=42x^5[/tex]
and solving for [tex]\dfrac{\mathrm d^2y}{\mathrm dx^2}[/tex] gives
[tex]\dfrac{\mathrm d^2y}{\mathrm dx^2}=\dfrac{42x^5-\frac{147x^{12}}{4y^4}}{4y^3}[/tex]
[tex]\dfrac{\mathrm d^2y}{\mathrm dx^2}=\dfrac{168x^5y^4-147x^{12}}{16y^7}[/tex]
