Derivatives and Their Approximation

Background The derivative of a function , is defined by where this limit exists. An examination of the graphs of for small nonzero values of h reveals some important relationships between a function and its derivative. It also shows that in many cases, the value of h doesn't have to be all that small before closely approximates .

Instructions Tips Enter the function, using independent variable x: Enter the left and right endpoints of the interval you would like view. If the right endpoint isn't greater than the left, you will get . left: right: Enter a nonzero value for h: To suppress the approximate derivative, enter 0. Select an "Aspect Ratio." You will usually get the nicest graphs when you select "1/GoldenRatio." If you select "Automatic" a uniform scale will have the physical slopes of tangent lines match the derivative. 1/GoldenRatio Automatic

Ken Levasseur
Mathematical Sciences
UMass Lowell
Kenneth_Levasseur@uml.edu
Ken's WebMathematica Scripts

Tips on using this script

1. You can enter any numeric function using Mathematica notation in Step 1 of the instructions - see examples below
2. Some functions like the absolute value function, Abs[x], have a derivative that is non-numeric -- D[Abs[x],x] evaluates to Abs'[x]. In these cases, no derivative is plotted.
3. You can enter valid numeric expressions using Mathematica syntax for the endpoints in Step 2, including expressions like 10^2, 5!, Sqrt[2], E, or Sum[3^k,{k,0,4}]
4. If you don't want to see the approximate derivative, enter zero for the value of h in Step 3.
5. Some interesting examples for Step 1
   • Binomial[x,k], where k is a positive integer.
   • Sum[x^k/k!,{k,0,5}]
   • ChebyshevT[3,x], or change the 3 to some other positive integer.
   • An off-the-wall example: Random[Real,{0,x}]