$X$ and $Y$ are smooth manifolds and let $f:X\to Y$ be a map. Let $\Gamma$ be the graph of $f$ in $X\times Y$. Prove that $T_{(x,y)}\Gamma$ is the graph of $df(x):T_xX\to T_yY$.
$\Gamma$ need not be a smooth manifold, does it? If it were I could parametrize it and define $T_{(x,y)}\Gamma$ to be the image of a parametrization. I don't know how this would help, though.
How do I prove this statement?