I'm trying to solve Chapter 5, exercise 2 in Do Carmo's Riemannian Geometry, but I think I'm differentiating the $\exp$ map incorrectly. He defines $f(s,t)=\exp_{\lambda (s)}tW(s)$ such that $\lambda(s)$ is a curve with $\lambda(0)=\gamma(0),\lambda ' (0)=J(0)$ and $W(s)$ is a vector field along $\lambda$ such that $W(0)=\gamma ' (0), \frac{D}{ds}W(0)=\frac{D}{dt}J(0)$. $\gamma$ is a given geodesic and $J$ is a Jacobi Field along $\gamma$ and this is supposed to be the case where $J(0)\neq 0$. He then asks to verify that $\frac{\partial f}{\partial s}(0,0)=J(0)$, but I calculate the following:
$$\frac{\partial f}{\partial s}=d(\exp_{\lambda (s)})_{tW(s)}(t\frac{D}{ds}W(s))$$
then evaluating at $(0,0)$ I get
$$\frac{\partial f}{\partial s}(0,0)=d(\exp_{\lambda(0)})_{0}(0)=0$$
so I have calculated something wrong because I should be getting $J(0)$.
I also had difficulties initially dealing with differentiating a function like $f(s,t)$ you mentioned, which depends on the exponential map with variable base point. However, I learned that Jacobi fields are actually there to compute such a derivative.
Now back to your problem: If you want to evaluate $\frac{\partial}{\partial s}f$ at $s=t=0$, you can keep $t$ fixed. Notice that $$ f(s, 0) = \exp_{\lambda(s)} 0 = \lambda(s), $$ as exponential map of the zero vector is the base point. Differentiating this gives $$ \frac{\partial}{\partial s}f(0,0)= \frac{\partial}{\partial s} \biggr\rvert_{s=0}\lambda(s) = J(0).$$