Q:

What is the difference between |(d/dt)velocity| and (d/dt)|velocity| [Physical Interpretation]

According to me the modulus of $\frac {\text{d}v}{\text{d}t}$ gives the magnitude of acceleration without accounting for direction but I am stumbled on how to interpret $\left(\frac{\text{d}}{\text{d}t}\right)|velocity|$

A:

As you said, $|d{\bf v}/dt|$ is the magnitude of acceleration.
$d|{\bf v}|/dt$ is the rate of change of speed.
In general they are not the same.
For example, in uniform circular motion
$$\bigg|\frac{d{\bf v}}{dt}\bigg|=\frac{|{\bf v}|^2}{R}$$
while
$$\frac{d|{\bf v}|}{dt}=0$$

