Derivatives

Slope Function

The following construction lets you introduce the derivative as a slope function. The mathlet shows a moveable point A that lies on the function f(x) = sin(x). We created the tangent t for point A in respect to f as well as its slope m. Then we defined point M as M = (x(A), m). When you drag point A along the x-axis, the trace of point M shows the slope of function f for every x. This is the graph of our slope function. Students can now try to find an equation for this slope function so that its graph fits the trace of M. Doing so, they are actually looking for the derivative of f.

You could use this construction to foster discussions about possible equations for the derivative of functions like sin(x), cos(x), exp(x) and so on. By changing the equation of f in the algebra window, you can explore any of these functions with this single construction. GeoGebra also offers a way to get the solution: Type Derivative[f] into the input field and you will get the symbolic derivative of f.

Note, that you can again use the tools "Move drawing pad" as well as "Zoom in" and "Zoom out" to make important details of the construction visible in the graphics window of the applet above.

Curve Sketching

Besides examining the derivative of a function you can also use GeoGebra to make the traditional curve sketching of polynomial functions more interesting. In order to examine a variety of functions at once, we created sliders to represent the parameters a, b, c and d of a cubic polynomial function. Afterwards we defined the equation f(x) = a * x^3 + b * x^2 + c * x + d via the input field. By using the commands f'(x) = Derivative[f] GeoGebra showed us the equation and the graph of the first derivative of f . To locate all roots and all extrema we used Roots[f] and Extremum[f]. Modify the parameters of f in the construction below by dragging the sliders to examine different polynomials.

This construction lends itself to investigate the relation between the extrema of the function f and the roots of its derivative. Use the command Root[f'] in the input field to get these roots. In a next step, you could also create the second derivative of f with Derivative[f, 2] and analyze its course in respect to f and the first derivative. Note that you are still able to modify the parameters of the polynomial function f by dragging the sliders.