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I was just curious as to what "$'$" denotes; i.e. $x' = y$, as in $x'(t) = x(t)$ which has the solution $x(t) = c_1\;e^t$.

I've found out that it has something to do with differential equations, but I can't seem to find any information specifically on "$x'$".

If someone could provide a source to such information, it would be much appreciated.

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    $\begingroup$ needs more context $\endgroup$ – Jonathan Jun 4 '13 at 6:01
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    $\begingroup$ It is almost certainly the derivative of $x$ with respect to $t$. The Leibniz notation for this would be $\frac{dx}{dt}$. Added: The new material settles things: it is the derivative. $\endgroup$ – André Nicolas Jun 4 '13 at 6:05
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Typically, $x'$ refers to some derivative of $x$ with respect to a given variable. It is often used in contexts where the derivative being taken is clear, for ease of notation.

In the equation you list, for example, $x'(t) = x(t)$ is the same thing as writing $\frac{d}{dt}x(t) = x(t)$.

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    $\begingroup$ This use of primes is Lagrange notation; $dx/dt$ expresses the same thing as $x'(t)$, but $dx/dt$ is said to use Leibniz notation. $\endgroup$ – Muphrid Jun 4 '13 at 6:05
  • $\begingroup$ Thanks for answering! This cleared things up pretty quickly! Cheers! $\endgroup$ – JohnWO Jun 4 '13 at 6:09
  • $\begingroup$ @JohnWO, no problem, glad I could help! And Muphrid, thanks for the additional information. $\endgroup$ – Alex Wertheim Jun 4 '13 at 6:12
  • $\begingroup$ The prime notation is basically Newton's notation, but slightly changed (Newton wrote dots above the letter, for example $\ddot{x}=x^{\prime\prime}$). There was a bitter dispute between the followers of Newton and the followers of Liebniz as to who "invented" the calculus, and the followers of Newton (aka, mathematicians from Britain) refused to use Liebnitz notation, out of patriotism. This ended up isolating British mathematics from the rest of Europe, and they never caught up until Hardy at the start of the twentieth century. (cont.) $\endgroup$ – user1729 Jun 4 '13 at 10:04
  • $\begingroup$ It is interesting that it wasn't just the isolation which held them back, but the actual notation. Now, you may ask "why would this notation hold them back?" Well, try to write the chain rule using Newton's notation. It is not easy, is it... My point: notation is important. (Further reading) $\endgroup$ – user1729 Jun 4 '13 at 10:04
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It is a symbol of derivative, it means $\displaystyle \frac{dx}{dt} = y$. You can find the solution by separation of variables technique.

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