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What's the differentiation of $2x\left(\frac{dx}{dt}\right)$ with respect to the variable $t$?

Is it $$2\left(\frac{dx}{dt}\right)^2+2x\left(\frac{d^2x}{dt^2}\right)?$$

Asaf Karagila
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General Kenobi
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2 Answers2

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Let $g(t) = 2 x(t) \frac{dx}{dt}(t)$.

By the product rule for differentiation we get

$$ \frac{dg}{dt}(t) = 2 \left(\frac{dx}{dt}(t)\right)^2 + 2x(t)\frac{d^2x}{dt^2}(t) $$

so yes you are correct!

DodoDuQuercy
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    ooh I wish I was! I need this for a physics question and if I put the values I get a wrong answer! But thanks for the help @DodoDuQuercy – General Kenobi Jun 25 '20 at 11:25
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    Perhaps there is an error with the question? – ilsec Jun 25 '20 at 11:45
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    @GeneralKenobi, is this the entire question? Do the values you're putting in have something to do with another part of the question that you haven't shown us? – Izaak van Dongen Jun 25 '20 at 11:48
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    @IzaakvanDongen no this is not the entire question, and yes the values I'm putting in are related to another part of the question. If you wanna see that question, I've posted it on the physics stack exchange, and you can check it out there if you want. – General Kenobi Jun 25 '20 at 15:01
  • @GeneralKenobi : That offer would be substantially improved by linking to that question on Physics.SE... – Eric Towers Jun 25 '20 at 20:16
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Use the product rule:

$\frac{d}{dt}(2x⋅\frac{dx}{dt})=\frac{d}{dt}(2x)⋅\frac{dx}{dt}+2x⋅\frac{d}{dt}(\frac{dx}{dt})=2⋅\frac{dx}{dt}⋅\frac{dx}{dt}+2x⋅\frac{d^2x}{dt^2}=2(\frac{dx}{dt})^2+2x⋅\frac{d^2x}{dt^2}$

This seems to be the answer, as @DodoDuQuercy pointed out. Perhaps there is a mistake in your assignment?

ilsec
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