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I'm following Professor R. Shankar's Fundamentals of Physics course on YouTube.

There I saw him doing manipulations of Calculus I never saw before.

Here it goes, $$\newcommand\deriv[2]{\frac{\mathrm d #1}{\mathrm d #2}} \deriv{v_t}{t}=a$$ $$\implies v_t \newcommand\deriv[2]{\frac{\mathrm d #1}{\mathrm d #2}} \deriv{v_t}{t}=a v_t$$ $$\implies \newcommand\deriv[2]{\frac{\mathrm d #1}{\mathrm d #2}} \deriv{\dfrac{{v_t}^2}{2}}{t}=a \newcommand\deriv[2]{\frac{\mathrm d #1}{\mathrm d #2}} \deriv{x_t}{t}$$

Then he cancels out $\mathrm dt$ from both sides and obtains, $\mathrm d(\dfrac{{v_t}^2}{2})=a\cdot \mathrm d(x_t)$

And proceeds by, $\dfrac{1}{2} \mathrm d({v_t}^2)=a\cdot \mathrm d(x_t)$

Thus deriving, $\dfrac{1}{2} ({v_t}^2-{v_0}^2)=a (x_t-x_0)$ $\implies {v_t}^2-{v_0}^2=2a(x_t-x_0)$

So, ${v_t}^2={v_0}^2+2a(x_t-x_0)$

How do you justify his calculations using Mathematical rigour?

Is it legitimate to cancel out $\mathrm dt$ from both sides? Aren't we considering $\newcommand\deriv[2]{\frac{\mathrm d #1}{\mathrm d #2}} \deriv{v_t}{t}$ as a fraction then? And Isn't there a better way of deriving this using Integral calculus?

Qmechanic
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ARahman
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    Related: https://physics.stackexchange.com/q/92925/2451 , https://physics.stackexchange.com/q/70376/2451 and links therein. – Qmechanic Oct 07 '18 at 08:58
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    Possible duplicate of How to treat differentials and infinitesimals? – John Rennie Oct 07 '18 at 09:06
  • @JohnRennie Your comment is duplicate. Qmechanic already gave the link you mentioned. – ARahman Oct 07 '18 at 09:19
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    @ARahman That's probably because JR's comment was auto-generated with the duplicate vote. QMechanic didn't vote to officially mark them as duplicates. –  Oct 07 '18 at 10:02
  • @Chair I see QMechanic is good guy ; not a police. – ARahman Oct 07 '18 at 10:05
  • @ARahman It sounds like you're passively attacking other users. What's likely is that QMechanic chose not to officially mark them as dupes because as a moderator, his vote would be instantaneous and unilateral. S/he decided to leave those links there and let the community decide whether or not they're duplicates. After a few of us expressed that it's probably a duplicate, Qmechanic sped things up by finishing off the review process: you'll see that only 4 of us voted to mark it as a duplicate, though usually it needs 5 voters. –  Oct 07 '18 at 10:09

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I had the same doubt(can we cancel dt or any differential).I asked in a math forum.They told me that for basic calculations these can be be cancelled.But another interpretation is, if the rate of change two things with respect to time are propotional or equal then the change in their value for a given period of time will also be equal or propotional.

Mohan
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