Applet: Ordinary derivative by limit definition
A function $g(x)$ is plotted with a thick green curve. The point $(a,g(a))$ (i.e., the point on the curve with $x=a$) is plotted as a large black point, which you can change with your mouse. The smaller red point shows the point on the curve with $x=a+h$, where you can change $h$ by dragging the blue point on the slider with your mouse. The blue line through the black and red points has slope given by \begin{align} \frac{g(a+h)-g(a)}{h}. \end{align}
Can you see why this is true? Since the height of the red point is $g(a+h)$ and the height of the black point is $g(a)$, the “rise” is $g(a+h)-g(a)$. The “run” between the points is $h$. So, rise over run is given by the above equation.
As you decrease $h$ toward zero, this slope of the blue line approaches the derivative $g'(a)$, as the above expression in this limit is exactly the limit definition of the derivative \begin{align} g'(a) = \lim_{h \to 0}\frac{g(a+h)-g(a)}{h}. \end{align}
Applet file: ordderiv.m
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