What does an apostrophe mean in a function?

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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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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.

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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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