Estimate the gradient of a function on a complete manifold

Question

There is a function $f$ on the smooth, complete manifold with $f(x_{t})=f(x)+td(x,y)$, $t \in [0,1]$ where $x$ and $y$ are fixed points on the manifold, $d(x,y)$ is the geodesic distance, $x_{t} = {\rm{exp}}_{x}(t \log_{x}(y))$. My question is: whether ${\rm{grad}} \, f(x_{t}) = \frac{\log_{x_{t}}(y)}{d(x_{t},y)}$, Or whether we can get $|| {\rm{grad}} \, f(x_{t}) ||_{x_{t}}$ ?