Let $M$ be a smooth manifold and $f: I \to M$ be differentiable, where $I \subseteq \mathbb{R}$ denotes an interval. I have seen the notation $\frac{d}{dt}$ used in this context, for example $\left.\frac{d}{dt}\right\vert_{t=0} f$. I wondered what is meant by this notation. Does it mean $\partial_f,$ where $$\partial: T_{f(0)}M \to \text{Der}_{f(0)}(M), \ \partial_{f}([g]_{f(0)})=(g \circ f)'(0),$$ where on the right hand side the "common" derivative is meant? Are there more general settings where a similar notation is being used, for example for maps $f:M \to N$ between two smooth manifolds?
Edit for clarification: $T_pM$ denotes the geometric tangent space of $M$ at a point $p$ and $\text{Der}_pM(M)$ denotes the algebraic tangent space or the space of derivations in $p$.The function $\partial$ assigns to every tangent vector $[f]_p$ the function (derivation) $\partial_f$ as given above.
An example, which I just stumbled upon, can be found in the book Riemannian Manifolds: An introduction to curvature by John M. Lee on page 76 in the proof of lemma 5.10: $$(\text{exp}_p)_*V=\left.\frac{d}{dt}\right\vert_{t=0}(\text{exp}_p \circ \tau)(t)(=…)$$
Best Answer
The interval $I$ is a one-dimensional manifold that has the vector field $\frac{\partial}{\partial t}\,.$ A functon $f:I\to M$ is a curve in the manifold $M$ and it is common to define $f'(t)$ to be the map that maps $\frac{\partial}{\partial t}$ to the tangent vectors at this curve. To do so we need real valued test functions $\varphi$ defined on $M$ so that we can traditionally differentiate $\varphi\circ f$: $$\tag{1} df_t\Big(\frac{\partial}{\partial t}\Big)(\varphi)=(\varphi\circ f)'\,. $$ Many authors now write $$ f'(t)=df_t\Big(\frac{\partial}{\partial t}\Big)\in T_{f(t)}M\,. $$ On p. 56 John M. Lee writes (1) as $$ \dot f(t)\,\varphi=\frac{d}{dt}(\varphi\circ f)(t). $$ (I used $f$ for his $\gamma$ to avoid confusion). A reasonable interpretation for $\frac{d}{dt}\Big|_{t=0}f$ would now be $f'(0)$ i.e. the map that sends $\frac{\partial}{\partial t}\Big|_{t=0}$ to the tangent vector in $T_{f(0)}M\,.$
John M. Lee recommends -in a personal comment for which I am very grateful- his book Introduction to Smooth Manifolds (2nd ed.), Chapter 3, especially pp. 68-70.