Short proof of vector function derivative

[Unununium111]

Hey there,
So I find myself struggling with another problem. I sometimes cannot visualize the problem when in the form "prove this" is equal to some arbitrary, intermediate equation proof....which is exactly what this is. I'm sure the answer is right in front of me, I just wish it had real numbers. Let me know what you think...

If |r(t)|>0 , then show that (d/dt)|r(t)| = {(1/|r(t)|)*r(t)} * ((d/dt)r(t))

What happens when |r(t)| --> 0 with d/dt{r(t)} ?

Thanks in advance!!

 

[galactus]

I assume you mean,

Let r=r(t), and

Prove \(\displaystyle \frac{d}{dt}[||r||]=\frac{1}{||r||}r\cdot r'\)

If so, consider \(\displaystyle r\cdot r\)

\(\displaystyle ||r||^{2}=r\cdot r\)

Thus, \(\displaystyle \frac{d}{dt}||r||^{2}=\frac{d}{dt}(r\cdot r)\)

\(\displaystyle 2||r||\frac{d}{dt}(||r||)=r\cdot \frac{dr}{dt}+\frac{dr}{dt}\cdot r=2r\cdot r'\)

\(\displaystyle \frac{d}{dt}(||r||)=\frac{r\cdot r'}{||r||}\)