Use the sliders or input boxes to set the coefficients of the original cubic polynomial function. Check the checkbox for f(x) to see its graph in blue. Notice that the domain and range are both the set of all real numbers. The function is continuous and smooth. The graph of the original function touches the [i]x[/i]-axis 1, 2, or 3 times. These [i]x[/i]-intercepts are indicated. It crosses the y-axis exactly once. This [i]y[/i]-intercept is indicated on the graph.

Check the checkbox on [i]f[/i] '([i]x[/i]) to see the formula and graph of the first derivative of the original function in red. Notice that it is a quadratic (second degree) polynomial function. Notice that when the first derivative is positive (above the [i]x[/i]-axis), the original function is increasing. When the first derivative is negative (below the [i]x[/i]-axis), the original function is decreasing. When the first derivative is zero (on the [i]x[/i]-axis) and the second derivative is not zero, the original function has local extrema. The original function will either have exactly one local maximum and one local minimum or it will have no extrema. The extrema are indicated on the original graph.

The second derivative of the original function is a linear (first degree polynomial) function. It will always intersect the [i]x[/i]-axis exactly once. This is the same [i]x[/i]-value where the first derivative has an extremum and the original function has an inflection point. These points always exist, and they are indicated on the respective graphs.[br]

The third derivative is a constant function. The fourth and all higher order derivatives are the zero function ([i]x[/i]-axis). Click the checkbox for General Derivative Formulas to see the general formulas for all derivatives in terms of the parameters [i]a, b, c[/i], and [i]d[/i]. Check the Proof checkbox to see a proof/derivation of the first derivative formula working directly from the definition of the derivative.