Conjugate locus and local diffeomorphism problem in do Carmo's book.

Question

I'm reading DoCarmo's book, Riemannian Geometry and i don't understand something. At page 149 first lemma

My question is why $\text{exp}_p$ is local diffeomorphism?

Answer 1

For $p\in M$ identify $T_{0}T_{p}M$ with $T_{p}M$ then :

$d(\exp_{p})_{0}(X_{p})=\frac{d}{dt}_{t=0}(\exp_{p}(tX_{p}))=X_{p}$ , $X_{p}\in T_{p}M$ .

the rest concluded by inverse function theorem.