\frac{\partial^2 y}{\partial \alpha^2} = \frac{\partial}{\partial \alpha} \Big{(} \frac{\partial y}{\partial \alpha} \Big{)} = \frac{\partial}{\partial \alpha} \Big{(} f(y) \frac{\partial y}{\partial x} \Big{)} = f(y) \frac{\partial}{\partial x} \Big{(} f(y) \frac{\partial y}{\partial x}\Big{)} = \frac{\partial}{\partial x} \Big{(} f^2(y) \frac{\partial y}{\partial x}\Big{)}.

Par récurrence on a bien \displaystyle \frac{\partial^n y}{\partial \alpha^n} = \frac{\partial^{n-1}}{\partial x^{n-1}} \Big{(} f^n(y). \frac{\partial y}{\partial x}\Big{)}.