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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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.
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)