Derivative related to equal areas proof

[mishima]

Kleppner and Kolenkow's Introduction to Mechanics text on page 241 (1st edition) has:

$\frac{d}{dt}$(r²$\dot{\theta}$)=r(2$\dot{r}$$\theta$+r$\ddot{\theta}$)

Is this wrong or am I missing something? I get:

r(2$\dot{\theta}$+r$\ddot{\theta}$)

...by product rule. It seems at the least the book should have a theta dot in the first term. I don't see where the r dot comes from though. Thank you.

 

[PAllen]

Kleppner and Kolenkow's Introduction to Mechanics text on page 241 (1st edition) has:

$\frac{d}{dt}$(r²$\dot{\theta}$)=r(2$\dot{r}$$\theta$+r$\ddot{\theta}$)

Is this wrong or am I missing something? I get:

r(2$\dot{\theta}$+r$\ddot{\theta}$)

...by product rule. It seems at the least the book should have a theta dot in the first term. I don't see where the r dot comes from though. Thank you.

It does seem the book is missing a dot over the first theta on the right hand side. Otherwise the book answer is correct, and your is not. As for the r dot, what do you think is the result of:

d/dt (r^2)

 

[mishima]

$\frac{d}{dx} (f(x))^2 = 2f(x)\frac{d}{dx}f(x)$

Thanks, I was doing d/dr instead of d/dt. And thanks for confirming the theta dot.