Let us consider a point P in cylindrical co-ordinate system, having cylindrical co-ordinates (\rho,\ \phi,\ z) . The cartesian co-ordinates of the point P is (x,\ y,\ z) . Let’s draw a vertical line PQ on the X-Y plane, which is parallel to the z-axis, as shown in Fig.1.

OQ is represented by \rho , which makes an angle \phi with the X-axis.
From Fig. 1 we can write,
x=\rho\ \cos\phi
y=\rho\ \sin\phi
and, z=z
If \vec{r} is the position vector with respect to the origin O, then we can write
\vec{r}=x\ \hat{i}+y\ \hat{j}+z\ \hat{k}\tag{1}
where, \hat{i} , \hat{j} and \hat{k} is the unit vector along the X-axis, Y-axis and Z-axis respectively.
Now, equation (1) can be written as,
\vec{r}=\rho\ \cos\phi\ \hat{i}+\rho\ \sin\phi\ \hat{j}+z\ \hat{k}\tag{2}
The unit vectors \hat{\rho} , \hat{\phi} and \hat{z} in the cylindrical pola co-ordinate system in terms of the unit vecctors in certesian co-ordinate can be written as,
\hat{\rho}=\cos\phi\ \hat{i}+ \sin\phi\ \hat{j}\tag{i}
\hat{\phi}=-\sin\phi\ \hat{i}+\cos\phi\ \hat{j}\tag{ii}
and, \hat{z}=\hat{k}\tag{iii}
[ To know the derivation (CLICK HERE) ]
Again,
Differentiating \hat{\rho} with respect to time t we get,
\frac{d\hat{\rho}}{dt}=\frac{d}{dt}(\cos\phi\ \hat{i}+ \sin\phi\ \hat{j})
or,\ \frac{d\hat{\rho}}{dt}=\frac{d}{d\phi}(\cos\phi\ \hat{i}+ \sin\phi\ \hat{j})\frac{d\phi}{dt}
or,\ \frac{d\hat{\rho}}{dt}=(-\sin\phi\ \hat{i}+\cos\phi\ \hat{j})\dot{\phi}
or,\ \displaystyle{\frac{d\hat{\rho}}{dt}=\dot{\phi}\ \hat{\phi}}\tag{iv} using equation (ii)
Differentiating \hat{\phi} with respect to time t , we get
\frac{d\hat{\phi}}{dt}=\frac{d}{dt}(-\sin\phi\ \hat{i}+\cos\phi\ \hat{j})
or,\ \frac{d\hat{\phi}}{dt}=\frac{d}{d\phi}(-\sin\phi\ \hat{i}+\cos\phi\ \hat{j})\frac{d\phi}{dt}
or,\ \frac{d\hat{\phi}}{dt}=-(\cos\phi\ \hat{i}+\sin\phi\ \hat{j})\dot{\phi}
or,\ \displaystyle{\frac{d\hat{\phi}}{dt}=-\dot{\phi}\hat{\rho}}\tag{v} using equation (i).
Differentiating \hat{z} with respect to time t we get,
\displaystyle{\frac{d\hat{z}}{dt}=0}\tag{vi}
Multiplying both side of equation (i) by \cos\phi and equation (ii) by \sin\phi and then subtracting, we get
\hat{i}=\cos\phi\ \hat{\rho}-\sin\phi\ \hat{\phi}\tag{vii}
Multiplying both side of equation (i) by \sin\phi and equation (ii) by \cos\phi and then adding, we get
\hat{j}=\sin\phi\ \hat{\rho}+\cos\phi\ \hat{\phi}\tag{viii}
Now putting value of \hat{i} , \hat{j} and \hat{k} in the equation (2), by using equaions (vii), (viii) & (iii), we get
\vec{r}=\rho\ \cos\phi\ (\cos\phi\ \hat{\rho}-\sin\phi\ \hat{\phi})+\rho\ \sin\phi\ (\sin\phi\ \hat{\rho}+\cos\phi\ \hat{\phi})+z\ \hat{z}
or,\ \vec{r}=(\rho\ \sin^2\phi+\rho\ \cos^2\phi)\hat{\rho}+(\rho\ \cos\phi\ \sin\phi-\rho\ \cos\phi\ \sin\phi)\hat{\phi}+z\ \hat{z}
or,\ \vec{r}=\rho\ \hat{\rho}+z\ \hat{z}\tag{3}
Now eqation (3) can be written as,
\vec{r}=r_{\rho}\ \hat{\rho}+r_{\phi}\ \hat{\phi}+r_{z}\ \hat{z}\tag{4}
where, r_{\rho}=\rho , r_{\phi}=0 and r_{z}=z are the components of the position vectors \vec{r} along the direction of the unit vectors \hat{\rho} , \hat{\phi} and \hat{z} respectively.
Expression for velocity in cylindrical polar co-ordinate system:
Since velocity is the time derivative of the displacement vector,
So, velocity \displaystyle{\vec{v}=\frac{d\vec{r}}{dt}}
Now differentiating equation (4) with respect to time t, we get
\displaystyle{\frac{d\vec{r}}{dt}=\frac{dr_{\rho}}{dt}\ \hat{\rho}+r_{\rho}\frac{d\hat{\rho}}{dt}+\frac{dr_{\phi}}{dt}\hat{\phi}+r_{\phi}\frac{d\hat{\phi}}{dt}+\frac{dr_{z}}{dt}\hat{z}+r_{z}\frac{d\hat{z}}{dt}}
or,\ \displaystyle{\frac{d\vec{r}}{dt}=\frac{dr_{\rho}}{dt}\ \hat{\rho}+r_{\rho}\ \dot{\phi}\ \hat{\phi}+\frac{dr_{\phi}}{dt}\hat{\phi}+r_{\phi}(-\dot{\phi}\hat{\rho})+\frac{dr_{z}}{dt}\hat{z}}
[ using equations (iv), (v) & (vi) ]
or,\ \displaystyle{\frac{d\vec{r}}{dt}=\frac{d\rho}{dt}\ \hat{\rho}+\rho\ \dot{\phi}\ \hat{\phi}+\dot{z}\ \hat{z}}
or,\ \displaystyle{\frac{d\vec{r}}{dt}=\dot{\rho}\ \hat{\rho}+\rho\ \dot{\phi}\ \hat{\phi}+\dot{z}\ \hat{z}}\tag{5}
So equation (5) is the expression for the velocity of a particle in a cylindrical polar co-ordinate system.