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