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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4$\begingroup$ It's a very common notation for the derivative. $\endgroup$– Git GudCommented Jun 5, 2015 at 13:43
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$\begingroup$ $f'(x)=\frac{df}{dx}$ $\endgroup$– JJacquelinCommented Jun 5, 2015 at 13:45
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7$\begingroup$ 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? $\endgroup$– Au101Commented Jun 5, 2015 at 13:51
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1$\begingroup$ I suppose we need the ask the OP do they know what the derivative is? $\endgroup$– JP McCarthyCommented Jun 5, 2015 at 14:00
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$\begingroup$ @Au101 As an American, I prefer the "dash" enunciation, but you're right that "prime" is very common here. $\endgroup$– KenCommented Jun 5, 2015 at 14:02
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2 Answers
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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.
answered Jun 5, 2015 at 13:56
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1$\begingroup$ "It means the first derivation of the function...": it's "derivative", not "derivation". $\endgroup$– TonyKCommented Jun 5, 2015 at 14:02
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$\begingroup$ My bad, thank you for the correction :) $\endgroup$ Commented Jun 5, 2015 at 14:02
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$\begingroup$ I've seen $f'''(x)$. 4 primes is pushing it. $\endgroup$ Commented Jun 3, 2017 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)
answered Jun 3, 2017 at 3:58