I'm working on a IVP $$ y'(t) = f(t,y), \\ y(t_0) = y_0 $$ and am struggling with the derivatives (I need them for a Taylor expansion). I hope this isn't a dumb question , but can someone tell me if the following is correct? That would be very helpful!
$$ \frac{d^3y}{dt^3} = y'''(t) = \frac{d^2f}{dt^2} =\\ = \frac{\partial^2f}{\partial t^2} + 2 \cdot \frac{\partial^2f}{\partial y\partial t} \cdot f + \frac{\partial^2f}{\partial y^2} \cdot f^2 + \frac{\partial f}{\partial y} \cdot \frac{\partial f}{\partial t} + \left( \frac{\partial f}{\partial y} \right)^2 \cdot f $$