How can I compute the differential of a projective morphism?

Question

I want to understand how to compute the differential of a projective morphism of schemes. For example, consider the projective morphism $$ \begin{matrix} & \textbf{Proj} \left(\frac{\mathbb{C}[s,t][x,y,z]}{x^3 + y^3 + s(x^2y + z^3) + t(xyz)} \right) \\ & \downarrow \pi \\ & \textbf{Spec}(\mathbb{C}[s,t]) \end{matrix} $$