Difference between derivatives and limits? Hey I have a small question - as I am getting into derivatives, I was shown how to take the limit as $h$ approaches $0$ and to create an equation. Additionally, we were told to never forget using the limit.
Then how come when we just have a function and we take the derivative of that function, we don‘t use limits?
Thanks in advance hopefully I could explain more or less what I mean....

The derivative is by definition the limit $$f'(x) = \lim_{h\rightarrow 0} \frac{f(x+h) - f(x)}{h}.$$ So for example, if $f(x) = x^2$ then $f'(x) = 2x$ means exactly that $$\lim_{h\rightarrow 0} \frac{f(x+h) - f(x)}{h} = 2x.$$  You wouldn't write something like $$\lim_{h\rightarrow 0} f'(x) = 2x$$ because that would mean $$\lim_{h\rightarrow 0}  \lim_{h\rightarrow 0} \frac{f(x+h) - f(x)}{h} = 2x$$ which doesn't make sense.
To prove that $f'(x) = 2x$ you do need to show that $$\lim_{h\rightarrow 0} \frac{f(x+h) - f(x)}{h} = 2x.$$ But once you know this, you don't have to write out the limit every time.  That's the power of notation.
