Let $X$ and $Y$ be algebraic varieties over an algebraically closed field $k$. Consider two points $a\in X$ and $b\in X$. I want to prove that the natural projection maps $p_X:X\times Y\to X$ and $p_Y:X\times Y\to Y$ induce an isomorphism of $k$-vector spaces $$ T_{(a,b)}(X\times Y)\to T_aX\oplus T_b Y, $$ where $T_zZ$ denotes the tangent space at a point $z$ in an algebraic variety $Z$. Since the notion of tangent space is local, we can assume that $X$ and $Y$ are affine algebraic varieties.
The notion of tangent space I'm using is through derivations. That is an element of $T_a X$ is a $k$-linear map $$ D:\mathcal{O}_{X,a}\to k $$ satisfying $$ D(fg)=D(f)g(a) + f(a)D(g). $$ For a morphism $f:X_1\to X_2$ of algebraic varieties and a point $x_1\in X_1$ and $x_2=f(x_2)$ we write $d_{x_1}f:T_{x_1}X_1\to T_{x_2}X_2$ for the differential of $f$ at $x_1$.
Now, here is my attempt, following a reasoning similar to the differentiable case: Define $$ \varphi:T_{(a,b)}(X\times Y)\to T_a X \oplus T_b Y $$ by $$ \varphi(D) = (d_{(a,b)}p_X(D),d_{(a,b)}p_X(D)). $$ This map is clearly linear. Now, let $i_b:X\to X\times Y$ and $j_a:Y\to X\times Y$ be the maps $$ i_b(x) = (x,b) \quad \text{and} \quad j_a(y)=(a,y). $$ This are clearly morphism of affine varieties. Thus define $$ \psi:T_a X\oplus T_b Y\to T_{(a,b)} (X\times Y) $$ by the formula $$ \psi(D_1,D_2) = d_a i_b(D_1) + d_b j_a(D_2). $$ The aim is to show that $\psi$ is a two-sided inverse for $\varphi$. The composition $\varphi\circ \psi$ is easy to evaluate, but I'm stuck computing the composition $\psi\circ \varphi$. To be more precise, one has, for $f\in \mathcal{O}_{X\times Y,(a,b)}$ and $D\in T_{(a,b)}(X\times Y)$ $$ \psi(\varphi(D))(f) = D(f\circ i_b\circ p_X + f\circ j_a\circ p_Y), $$ and I'll be done if I prove that $f\circ i_b\circ p_X + f\circ j_a\circ p_Y - f-f(a,b)\in \mathfrak{m}_{(a,b)}^2$, where $\mathfrak{m}_{(a,b)}$ is the unique maximal ideal of the local ring $\mathcal{O}_{X\times Y,(a,b)}$. Here is where I'm stuck (I don't even know if this is true).
Any help will be appreciated.