Derivative

The derivative of a function (say $f(x)$) is a function of the slope of the tangent line at any given point. It is a generic slope formula for any x value, or instantaneous rate of change. The output of the derivative can be used to make tangent lines. Its notation is $f'(x)$ or $\frac{d}{dx} f(x)$.

Using the slope formula $m = \frac{y_2-y_1}{x_2-x_1}$, you can derive the definition of a derivative using a limit.

$$ \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} = f'(x) $$

The idea is to use a limit to get the instantaneous rate of change of a function as $h$ or the difference between $x_2$ and $x_1$ becomes indefinitely small. Basically get the slope from two points, but the two points are indefinitely close to eachother, or basically one point, which is the slope of a tangent line!

alternative form! and add a graphic for the definition of a derivative.