Let $M$ $n$-dimensional Riemannian manifold, $x\in M$ and $\phi\in C^{2}(M,\mathbb{R})$. Consider the function $f(v)=\phi(\exp_{x}(v))$ defined on a neighborhood of $0_x$ in $TM_x$.
Show that, $$f(v)=f(0)+\langle df(0),v\rangle_{x}+\frac{1}{2}\langle d^{2}f(0)v,v\rangle_{x}+o(\|v\|^{2})\qquad v\in TM_x$$
Clearly, since $\phi\in C^{2}(M,\mathbb{R})$, I can develop a Taylor series. So, if we define $g:\mathbb{R}\to\mathbb{R}$, $g(t)=f(tv)$ we have $$g(t)=g(0)+g'(0)t+\frac{1}{2}g''(0)t^2$$
Furthermore, $g'(t)=\frac{d}{dt}(g(t))=\triangledown f(tv)\cdot v$, (another question I have is, is true that $\triangledown f(tv)\cdot v=\langle df(t),v\rangle_{x}$?). Therefore, $$f(tv)=f(0)+\langle df(0),v\rangle_{x}t+\frac{1}{2}\langle d^2f(0)v,v\rangle_{x}t^2$$
Anyway, how can prove rigoursly way that $g''(t)=\triangledown[\triangledown f(tv)\cdot v]=[\triangledown^2 f(tv)\cdot v]\cdot v=\langle d^2f(tv)v,v\rangle_{x}$? Thanks!