Let ƒ(x) be a real differentiable function such that $\mathrm{f}\left(0\right)=1\text{ and }\mathrm{f}(\mathrm{x}+\mathrm{y})=\mathrm{f}\left(\mathrm{x}\right){\mathrm{f}}^{\mathrm{\prime }}\left(\mathrm{y}\right)+{\mathrm{f}}^{\mathrm{\prime }}\left(\mathrm{x}\right)\mathrm{f}\left(\mathrm{y}\right)$ for all $\mathrm{x},\mathrm{y}\in \mathbf{R}\text{. Then }\sum _{\mathrm{n}=1}^{100} {\log }_{\mathrm{c}}\mathrm{f}\left(\mathrm{n}\right)$ is equal to :

$\mathrm{f}(\mathrm{x}+\mathrm{y})=\mathrm{f}\left(\mathrm{x}\right){\mathrm{f}}^{\mathrm{\prime }}\left(\mathrm{y}\right)+{\mathrm{f}}^{\mathrm{\prime }}\left(\mathrm{x}\right)\mathrm{f}\left(\mathrm{y}\right)$

Put x=y=0

$\begin{array}{r}f\left(0\right)=f\left(0\right)f^{\prime }\left(0\right)+f^{\prime }\left(0\right)f\left(0\right)\end{array}$

$f^{\prime }\left(0\right)=\frac{1}{2}$

Put y=0

$\begin{array}{r}f\left(x\right)=f\left(x\right)f^{\prime }\left(0\right)+f^{\prime }\left(x\right)f\left(0\right)\end{array}$

$\begin{array}{r}f\left(x\right)=\frac{1}{2}f\left(x\right)+f^{\prime }\left(x\right)\end{array}$

$f^{\prime }\left(x\right)=\frac{f\left(x\right)}{2}$

$\begin{array}{r}\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{y}}{2}\Rightarrow \int \frac{\mathrm{dy}}{\mathrm{y}}=\int \frac{\mathrm{dx}}{2}\end{array}$

$\Rightarrow \text{ lny }=\frac{\mathrm{x}}{2}+\mathrm{c}$

$\begin{array}{r}\because f\left(0\right)=1\Rightarrow C=0\end{array}$

$\begin{array}{r}\ln y=\frac{\pi }{2}\Rightarrow f\left(x\right)=e^{x/2}\end{array}$

$\begin{array}{r}\ln f\left(n\right)=\frac{n}{2}\end{array}$

$\begin{array}{r}\sum _{n=1}^{100} \ln f\left(n\right)=\frac{1}{2}\sum _{n=1}^{100} n=\frac{5050}{2}\end{array}$

$=2525$