I have the sphere manifold $M$ with dimension n and a function $\hat{f}_x : T_x M \rightarrow \mathbb{R}$, which is defined by $\hat{f}_x (v) = f ( exp_x(v))$, where exp is the exponential map from $T_x M$ to $M$ at a point $x \in M$. $T_x M$ is the tangent space at a point $x.$
$f(y)$ is defined by $y^T A y$ for a positive definite symmetric matrix (so $f: M \rightarrow \mathbb{R}$) and a point $y \in M$, and the exponential map is given by $x * cos(\|v\|_2)+v*\frac{sin(\|v\|_2)}{\|v\|_2}$ for $x \in M, v \in T_x M$.
I am trying to find the derivative of $\hat{f}$ in terms of v. I tried deriving it by the common rules, but that doesn't work, since I need to take into account which space the functions are in. Can anybody tell me how to take on this problem?