I tried to solve exercise II.8.3 (a) in Hartshorne's Algebraic Geometry which reads as follows.
Let $X$ and $Y$ be schemes over another scheme $S$. Use (8.10) and (8.11) to show that $\Omega_{X\times_S Y/S} \cong p_1^*\Omega_{X/S} \oplus p_2^*\Omega_{Y/S}$.
First I observed that by (8.10) we know $\Omega_{X \times Y / Y} \cong p_1^* \Omega_{X / S}$. (8.11) gives an exact sequence $$p_2^*\Omega_{Y/S} \rightarrow \Omega_{X\times Y/S} \rightarrow \Omega_{X\times Y/Y} \rightarrow 0,$$ so both things together already look promising. We just have to show that the first arrow is injective and that the sequence splits.
My failed attempt to prove this follows.
Reading Matsumura's Commutative ring theory I noticed, that he already provides a criterion for exactly this (in the affine case). Given ring homomorphisms $k \rightarrow A \rightarrow B$ ($k$ is not necessarily a field), we get an exact sequence $$\Omega_{A/k} \otimes B \rightarrow \Omega_{B/k} \rightarrow \Omega_{B/A} \rightarrow 0,$$ which is exactly the sequence on affine opens. If $A \rightarrow B$ is 0-smooth (afaik this is also called formally smooth), the first arrow is injective and the sequence is in fact split-exact. So this looks exactly like what I want.
But I think this does not apply in our case, where $B = A \otimes_k C$ for some $k$-algebra $C$. Suppose $N \subset A$ is an ideal with $N^2 = 0$ and $C = A/N$. Then $B = A/N$ and the identity map $A \rightarrow A$ never admits a lift $A/N \rightarrow A$ if $N \neq 0$. So $B$ is in fact not smooth over $A$.