Computations on differential geometry

Question

I have to do some really long computations of basic riemannian geometry. I am trying to use Mathematica to check them. This is the kind of computations I want to do. Let $M$ be a riemannian manifold and $f:M\rightarrow M$ be given by $f(x)=x$, then I want a code that tells me that $$df_x(e_i)=e_i,$$ for $x\in M$ and $e_i\in T_xM$. Also some of the computations involve and inner product. For exampe, if $f:M\subset (\mathbb{R}^3,\langle, \rangle) \rightarrow \mathbb{R}$ is given by $f(x)=\langle x,x\rangle$ then I would like to have a code that tells me that $$df_x(e_i)=2\langle x,e_i\rangle, $$ for $x\in M$ and $e_i\in T_xM$. I would be thankful for any help.