derive expression to show that how drag force is dependent on velocity of moving object in fluid
There are many such expressions, depending upon the size, speed and shape of the object and the viscosity of the fluid.
A commonly used expression is:
Drag = (1/2)*(fluid density)*V^2*A*Cd
where A is the frontal cross section area facing the flow and Cd is a dimensionless constant (the "drag coefficient") that is usually obtained expermimentally.
Usually 0.1 < Cd < 1
With recent advances in computers and fluid dynamics software, Cd can be now calculated in many cases.
Cd depends upon two dimensionless numbers, the Reynolds number (rho*D*V/mu)and the Mach number (V/a).
rho = fluid density
mu = fluid viscosity
a = speed of sound.
At normal speeds of objects on the ground, the Mach number dependence can be ignored.