What is the difference between $dv/dt $ and $ dv/dt $ & between $dx/dt $ and $dx/dt $?
Solution 1
$\frac{dv}{dt}$ means the rate of change of the magnitude of velocity with respect to time. On the other hand $\frac{dv}{dt}$ means the magnitude of the rate of change of velocity with respect to time (i.e. the magnitude of acceleration).
Let's take an example: A body was travelling at 5m/s along the positive x  axis. Its velocity changed to 3m/s along ve x  axis in 1 second. Here $\frac{dv}{dt}$ is $\frac{35}{1} = 2m/s^2$ and $\frac{dv}{dt}$ is $\frac{35}{1}=8m/s^2$.
Solution 2
$v$ is velocity and $v$ is speed. So with $dv/dt$ you take the derivative of velocity and with $dv/dt$ you take it of speed.
Everytime there is a turn, velocity changes (because direction of motion changes) but it's magnitude (the speed) doesn't necessarily change as well.
 A satellite in constantspeed orbit will have a nonzero $dv/dt$ but a zero $dv/dt$.
 A car speeding up straight ahead has nonzero values of both.
Siddharth Garg
Updated on June 29, 2020Comments

Siddharth Garg over 3 years
Here, $v$ represents velocity vector and $a$ represents acceleration vector.

Steeven over 6 yearsHow can $\frac{dv}{dt}$ result in a negative value? The bars give the vector magnitude as an absolute value.