So let's say I have a function defined as $f(x) = e^{3x}$. Then it can be differentiated using particularly the chain rule to $e^{3x}\cdot 3$. And here comes the part that makes me wonder:
So the derivative of any function $g(x)$ is defined as $\lim_{h\to0}\frac{g(x + h)-g(x)}{h}$. Then, why can't I just calculate the derivative of $f(x)$ by $\lim_{h\to0}\frac{e^{3(x+h)}-e^{3x}}{h}$?
Instead, $\lim_{h\to0}\frac{e^{3(x+h)}-e^{3x}}{h} = e^{3x} \neq 3\cdot e^{3x}$. Why do I have to separately apply different rules, namely the chain rule in this case but possibly a lot more, instead of just being able to replace a function into $\lim_{h\to0}\frac{g(x + h)-g(x)}{h}$ and find its value as $h \to 0$ in "one step". Similarly, why can't you just calculate the limit of all functions in "one step" - it seems like you have to separately apply different rules (or limit laws).
In fact, we have that $$f'(x)=\lim_{h\to0} \frac{e^{3(x+h)}-e^{3x}}{h} = e^{3x}\cdot \underbrace{\lim_{h\to0} \frac{e^{3h}-1}{h}}_3=3e^{3x}.$$ And in fact, also, applying the different rules of derivation helps a lot to avoid you having to calculate limits that at first glance seem impossible. How do you compute the following limit? $$\lim_{h\to0} \frac{\arctan (x+h)-\arctan(x)}{h}$$