Difference between derivatives and limits?

Question

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....

Answer 1

By definition, a real function $f$ is differentiable at point $a\in R$ if and only if f is defined on some open interval I containing a and $\lim_{h\to0}\frac{f(a+h)-f(a)}{h}$ exists, in this case we call this limit the derivative of $f$ at $a$. So, the value of the derivative is a limit. The rules we use that apply to functions directly are just results that we have proven but everything goes back to the limit in the fundamental perspective. In addition, you could also see that there are some underlying assumptions that need to be satisfied for this definition to work ($f$ is defined on the open interval containing a, $a\in R$,etc..) This means when you apply the rules, you must also be aware of the assumptions, although definitions and rules could vary(consider one side derivatives), the fundamentals don't change.

Answer 2

The derivative of a function $f$ at $x$, defined as $f'(x)$ is supposed to provide the $ \lim_{h\rightarrow 0} \frac{f(x+h) - f(x)}{h}$. The limit is also supposed to exist however $h$ chooses to approach $0$. By the rules we derived, we are able to write down these limits. E.g. the chain rule.

Answer 3

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.