What does an apostrophe mean in a function? 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?

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.