# Definition of derivative of exponential map in Do Carmo's "Riemannian Geometry"

In the proof of Gauss' Lemma, Do carmo uses the notation $$(d exp_p)_v v$$ in Gauss' Lemma, but he doesn't seem to define it anywhere. If anyone could shed some insight, would be appreciated :)

The exponential map at $$p$$ is a map $$\exp_p:\Omega_p\to M$$ where $$\Omega_p$$ is some open subset of $$T_pM$$. Notice that $$T_pM$$ is a vector space, hence a smooth manifold, and $$\Omega_p$$ is an open subset, so it’s a smooth manifold in its own right. Next, $$M$$ is obviously a smooth manifold, and we have that $$\exp_p$$ is a map from the manifold $$\Omega_p$$ into the manifold $$M$$, and it is easy to prove smoothness.
So, for any ‘point’ $$v\in \Omega_p$$ we can consider the tangent map/differential/pushforward (there are a bunch of names…) which is a linear map $$d(\exp_p)_v:T_v(\Omega_p)\to T_{\exp_p(v)}M$$. So, for any tangent vector $$\xi\in T_v(\Omega_p)$$, we can consider the tangent vector $$d(\exp_p)_v(\xi)\in T_{\exp_p(v)}M$$.
Hopefully this all makes sense; I’m just carrying out the general definitions for $$f:N\to M$$ and $$df_p:T_pN\to T_{f(p)}M$$ for a point $$p\in N$$, and surely Do Carmo covers these definitions.
What’s special in this case is that $$T_pM$$ is a vector space, so $$\Omega_p$$ is an open set in a vector space, so for each ‘point’ $$v\in \Omega_p$$ there is a canonical isomorphism $$T_v(\Omega_p)\cong T_pM$$ (in general if $$V$$ is a vector space and $$\Omega\subset V$$ is open, then for each $$v\in\Omega$$, $$T_v\Omega\cong V$$). As a result (after composing by a suitable isomorphism on the domain) you can think of the differential of the exponential map as a linear map $$d(\exp_p)_v:T_pM\to T_{\exp_p(v)}M$$. So, for any vector $$w\in T_pM$$, I can produce the vector $$d(\exp_p)_v(w)\in T_{\exp_p(v)}M$$.
Finally, notice that in this special case, we have $$v\in \Omega_p\subset T_pM$$, so we can consider $$d(\exp_p)_v(v)\in T_{\exp_p(v)}M$$. Notice that the two $$v$$’s play different roles. The $$v$$ in the subscript is the point at which we’re taking the derivative, and the $$(v)$$ in parentheses refers to the tangent vector, and the linear map is being applied on this vector.