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The chain rule

  • Thread starter Niles
  • Start date
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1. 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:

[tex]
\frac{d}{dt}x^2 = \frac{d}{dx}x^2\frac{dx}{dt} = \frac{d}{dx}x^2\dot x.
[/tex]

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

danago

Gold Member
1,122
4
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.
 
Last edited:
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How is the proper way to write it then?

Thanks for replying.
 

danago

Gold Member
1,122
4
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:

[tex]

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

[/tex]
 

HallsofIvy

Science Advisor
Homework Helper
41,683
865
Better would be
[tex]\dot{x}\frac{dx^2}{dx}[/tex]
or just
[tex]\left(\frac{dx}{dt}\right)\left(\frac{dx^2}{dx}\right)[/tex]

Now, what is
[tex]\frac{d x^2}{dx}[/tex]?
 
32,580
4,310
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.
 
1,868
0
The reason why I didn't evaluate it in my first post was because I wasn't sure if I had to differentiate [itex]\dot x[/itex] as well, but now I would of course just write 2x instead.
 

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