Function derivatives

How to run a CDF Player simulation
download Functionderivatives.cdf
download Functionderivatives.nb (Mathematica notebook)

Notes:
Drag the blue marker along the x axis to change the function point for which the tangent line and its slope are calculated, The slope of the tangent line is the y-value of the function’s first derivative $f'\left( x \right)$ . If the second derivative check-box is selected you can see that the same relation between the function $f\left( x \right)$ and its first derivative $f'\left( x \right)$ also exist between the first derivative $f'\left( x \right)$ and the second derivative $f''\left( x \right)$ (see it yourself…).

Understanding

Ask yourself:

when the first derivative is positive the function is…
when the function is decreasing the first derivative is…
when the first derivative crosses the x axis the function has…
when the function crosses the x axis the first derivative is…
when the function has a local maximum (or minimum) the first derivative…
when the first derivative has a local maximum (or minimum) the second derivative…
when the function is concave upwards (smiling) its second derivative is…
when the function is concave downwards (sad) its second derivative is…
when the second derivative is zero the first derivative … and the function…
when the second derivative is positive (negative) the first derivative …. and the function….

[answers]

See also: “A poly function and its derivative“

Credits:
Derivative as a Function by Jim Brandt
Derivatives: A Look at Graphs by Abby Brown
from the Wolfram Demonstrations Project

Author: Luca Moroni – February 2014