kinematics - Is $left|frac{dvec{r}}{dt}right| = frac{d|vec{r}|}{dt};$?

Is magnitude of instantaneous velocity same as instantaneous speed?

More specifically, is $$\left|\frac{d\vec{r}}{dt}\right| = \frac{d|\vec{r}|}{dt}\; $$

Also Is it wrong to say that $\dfrac{d|\vec{v}|}{dt}$ is rate of change of speed?

Answer

Actually you're asking two different questions.

1. Is the magnitude of instantaneous velocity the same as instantaneous speed? Well, yes, that's the definition of instantaneous speed.


2. 
   

   Is this equation true? $$\biggl\lvert\frac{\mathrm{d}\vec{r}}{\mathrm{d}t}\biggr\rvert = \frac{\mathrm{d}\lvert\vec{r}\rvert}{\mathrm{d}t}$$ No, it's not - but instantaneous speed is the quantity on the left. The one on the right is the radial component of velocity in a circular coordinate system, and it is useful for some detailed calculations, but it's not one of the "basic" kinematic quantities (for most reasonable definitions of "basic").

   
   

   For fun: an example that shows the difference is uniform circular motion, where the quantity on the right is zero but the one on the left is not. Also note that the thing on the right can actually be negative, if the particle is getting closer to the origin over time.

   
   

Since $\vec{v} = \frac{\mathrm{d}\vec{r}}{\mathrm{d}t}$ by definition, the quantity on the left in the above equation is $\lvert\vec{v}\rvert$, and so $\frac{\mathrm{d}\lvert\vec{v}\rvert}{\mathrm{d}t}$ is the rate of change of speed.