Differentiating x2 w.r.t Time: Must I Differentiate \dot x?

[Niles]

Homework Statement

Hi all.

I have a time-dependent variable called x. I wish to differentiate x² with respect to the time t, and this is what I have done:

$$\frac{d}{dt}x^2 = \frac{d}{dx}x^2\frac{dx}{dt} = \frac{d}{dx}x^2\dot x.$$

where the dot over x denotes differentiation with respect to time t. Now my question is, must I differentiate $\dot x$ also with respect to x, since it is standing on its right side?

 

[danago]

Nah you dont. It is a little bit ambiguous how you have written it, but in this context it is just d(x^2)/dx MULTIPLED by x dot.

 

[Niles]

How is the proper way to write it then?

Thanks for replying.

 

[danago]

Its not that how you have written it is wrong, it is just that it could possibly be misinterpreted. I personally would have witten it with the x dot on the other side of the derivative operator just so there will be no confusion:

$$\dot x \frac{d}{dx} x^2$$

 

[HallsofIvy]

Better would be
$$\dot{x}\frac{dx^2}{dx}$$
or just
$$\left(\frac{dx}{dt}\right)\left(\frac{dx^2}{dx}\right)$$

Now, what is
$$\frac{d x^2}{dx}$$?

 

[Niles]

Now, what is
$$\frac{d x^2}{dx}$$?

Ahh, now you are just teasing me

Thanks to all for helping.

 

[Mark44]

I don't think Halls was teasing - you shouldn't leave it as dx^2/dx. I think you understand, but I'm not 100% certain.

 

[Niles]

The reason why I didn't evaluate it in my first post was because I wasn't sure if I had to differentiate $\dot x$ as well, but now I would of course just write 2x instead.