Aplety

Derivative

The derivative of the function $f:\mathbb{R} \to \mathbb{R}$ at the point $a$ is the limit $$ f'(a) = \lim_{h\to 0} \frac{f(a+h)-f(a)}{h} ~,$$if this limit exists. If the limit exists, then f is differentiable at $a$. The natural geometric interpretation of the derivative of $f$ at the point $a$ is the slope of the tangent line to the graph of $f$ at point $(a,f(a))$.

The following applet contains the graph of $f(x) = x ^ 3$. Drag the black circle to change the position of the point $(a, f(a))$ and Drag the red circle to change the position of the point $ (a + h, f (a + h)) $. Red color is is used for the tangent line to the graph of $f$ in a black point $(a,f(a))$ and the blue, dashed line is the straight line passing through the points $ (a, f(a)) $ and $(a + h,f(a+h))$.

Watch how the value of the quotient $\frac {\Delta y} {\Delta x} $ changes when the red dot approach to the black.