A body of mass 500   g moves along x -axis such that it's velocity varies with displacement x…

It is given that 

 $v=10\sqrt{x} \mathrm{m} {\mathrm{s}}^{-1}$

Force is defined by $F=m\frac{dv}{dt} ...\left(i\right)$

Multiplying and dividing equation (i) by $dx$

$$\begin{gathered} F=m\frac{dv}{dt}\frac{dx}{dx} \\ \Rightarrow F=m\frac{dx}{dt}\frac{dv}{dx}=mv\frac{dv}{dx} ...\left(ii\right) \end{gathered}$$

Finding the values of the terms in equation (ii)
$\frac{dv}{dx}=\frac{10}{2\sqrt{x}}=\frac{5}{\sqrt{x}}$
$v\frac{dv}{dx}=a=10\sqrt{x}\times \frac{5}{\sqrt{x}}=50 \mathrm{m} {\mathrm{s}}^{-2}$

Hence, the force is 
$F=\frac{1}{2}\times 50=25 \mathrm{N}$.