I'm going through Warner's book on differentiable manifolds. On page 8 he defines what it means for a map $f: U \subset M \to \mathbb R$ to be differentiable:
$f$ is differentiable iff $f \circ \psi$ is differentiable for all charts $\psi$ on $M$.
He does not proceed to give a definition of the derivative of $f$. I tried to do a web search but did not find a definition. Is the derivative of $f$ just defined to be the derivative of $f \circ \psi$?
What's the definition of the derivative of a map defined on manifolds?
Edit
At the bottom of page 105 in this book (in the proof of the regular level set theorem) the author calculates the Jacobian of a map $F: M \to N$. This Jacobian contains entries of the form ${\partial F \over \partial x_i}$. So, it seems to me that the derivative, at least partial derivatives exist (although a comment below by Mariano Suarez-Alvarez suggests otherwise)
Best Answer
Let $f:M\rightarrow \mathbb{R}$ be a function from a manifold M to the manifold $\mathbb{R}$. Also, let $\gamma$ be a smooth curve on the manifold $M$ with $\dot{\gamma}\in TM$ it's derivative.
Then, we define the derivative map of $f$ as the function that does the following \begin{equation} \begin{aligned} Df : TM & \rightarrow T\mathbb{R},\\ \dot{\gamma} & \mapsto df(\dot{\gamma})=\dot{\gamma}(f)= \dot{(f(\gamma))} \end{aligned} \end{equation}