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In a workbook, I saw the function $f(x)=x^2$. Then, there was the same function with an apostrophe $f'(x)$. It was stated that $f'(x)=2x$.

What is the apostrophe, and why does it change the function?

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    $\begingroup$ It's a very common notation for the derivative. $\endgroup$ – Git Gud Jun 5 '15 at 13:43
  • $\begingroup$ $f'(x)=\frac{df}{dx}$ $\endgroup$ – JJacquelin Jun 5 '15 at 13:45
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    $\begingroup$ Please note that this is actually a "prime" symbol, rather than an apostrophe. Here in Britain, we read this "$f$ dash(ed) of $x$", though I hear they do things rather differently in America. I think "$f$ prime of $x$" is fine? $\endgroup$ – Au101 Jun 5 '15 at 13:51
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    $\begingroup$ I suppose we need the ask the OP do they know what the derivative is? $\endgroup$ – JP McCarthy Jun 5 '15 at 14:00
  • $\begingroup$ @Au101 As an American, I prefer the "dash" enunciation, but you're right that "prime" is very common here. $\endgroup$ – Ken Jun 5 '15 at 14:02
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It means the first derivative of the function with respect to the variable $x$

So $f(x)=x^2$, $f'(x)=2x$ and even 2 can be used where you get $f''(x)=2$ because it's the second derivative, after that you don't use the primes anymore.

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    $\begingroup$ "It means the first derivation of the function...": it's "derivative", not "derivation". $\endgroup$ – TonyK Jun 5 '15 at 14:02
  • $\begingroup$ My bad, thank you for the correction :) $\endgroup$ – Zelos Malum Jun 5 '15 at 14:02
  • $\begingroup$ I've seen $f'''(x)$. 4 primes is pushing it. $\endgroup$ – eyeballfrog Jun 3 '17 at 4:16
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That is Lagrange's notation for derivatives.

One of the most common modern notations for differentiation is due to Joseph Louis Lagrange. In Lagrange's notation, a prime mark denotes a derivative. If $f$ is a function, then its derivative is written $f'(x)$
Notation for differentiation (Wikipedia)

In Lagrange's notation, the derivative of $f$ is expressed as $f'$ (pronounced "f prime").
Derivative notation review (Khan Academy)

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