We know that for any simply connected surface $M$,whose Gaussian curvature $K\leq 0$, for any $p\in M$, $exp\_p: T\_pM\to M$ is diffeomorphism.


We know that for any $v\in T\_pM$ and $w\in T\_v(T\_pM)\simeq T\_pM$, 
$$|d(exp\_p)\_vw|\geq |w|$$


That is we have $$\||d(exp\_p)\_v\|\_{op}\geq 1$$


Question: Can we find an upper bound of $\||d(exp\_p)\_v\|\_{op}$ for all $v\in T\_pM$ for particular $p$ in particular $M$. Or this is not atall possible for every $p$ and $M$.


If question is of level math.stackexchange, please suggest me. 



Upper bound of derivative of exponential map
To get an upper bound of the kind you seek in general, you need a lower bound on $K$. Thus, for example, if you know that $K\ge -c^2$ on your surface, then you get 
$$
\|\mathrm{d}(\exp\_p)\_v\|\_{op} \le \frac{\sinh\bigl(c\,|v|\bigr)}{c\,|v|}
\simeq 1 + \frac{\bigl(c\,|v|\bigr)^2}{3!} + \frac{\bigl(c\,|v|\bigr)^4}{5!} +\cdots.
$$
This follows by a standard application of Sturm comparison on geodesics leaving $p$, and is usually treated in a first course in differential geometry, at least at the graduate level. 


A more precise estimate is possible if you know that $-b^2\ge K\ge -c^2$ for some constants $0\le b\le c$, for then one has 
$$
\frac{\sinh\bigl(b\,|v|\bigr)}{b\,|v|}
\le\|\mathrm{d}(\exp\_p)\_v\|\_{op} \le \frac{\sinh\bigl(c\,|v|\bigr)}{c\,|v|}.
$$
Moreover, one knows that, in this case, $\mathrm{d}(\exp\_p)\_v$ has two singular values, $\sigma\_1\equiv1$ and $\sigma\_2(p,v) = \|\mathrm{d}(\exp\_p)\_v\|\_{op}\ge 1$. This is special for surfaces, though.


