The limit definition of a derivative for a function $f(x)$is as follows:
\[ \begin{align*} \lim_{h\to0} \dfrac{f(x+h) – f(x)}{h} \end{align*} \]
To understand this function, imagine two points. Both are on the $f(x)$ curve, and one is at $(x,f(x))$, while the other is at $(x+h,f(x+h)$. What this function says as the second point approaches the first point, the slope between them will be the derivative at $x$.
As an example, lets look at $f(x) = x^2 + 2x$. Using the definition, we get:
\[ \begin{align*} f'(x) &= \lim_{h\to0} \dfrac{f(x+h) – f(x)}{h}\\ \dfrac{d}{dx} (x^2 + 2x) &= \lim_{h\to0} \dfrac{(x + h)^2 + 2(x + h) – (x^2 + 2x)}{h}\\ &= \lim_{h\to0} \dfrac{x^2 + 2xh + h^2 + 2x + 2h – x^2 – 2x}{h}\\ &= \lim_{h\to0} \dfrac{h^2 + 2xh + 2h}{h}\\ &= \lim_{h\to0} h + 2x + 2 = 2x + 2 \end{align*} \]
We get that the derivative of $f(x) = x^2 + 2x$ is $\boxed{2x + 2}$.