Numeric Derivatives Using TI-83, TI-92

Date: 10/17/2003 at 09:41:27
From: Cindy
Subject: numeric derivatives

What is the difference between a derivative and a numeric derivative?  

When I needed to graph the sin (x) function and its derivative, I 
plugged sin(x) in for the original function, then I differentiated and 
got cos(x) for its derivative.  However, when I checked it on my 
calculator using the nDeriv (numeric derivative) function, it came up
as sin(2x)/2x, and obviously graphed differently than cos(x).

I am aware that the correct derivative of sin(x) is cos(x), I just 
don't understand why the numeric derivative gave a different answer.

Date: 10/17/2003 at 12:54:23
From: Doctor Peterson
Subject: Re: numeric derivatives

Hi, Cindy.

You didn't say what kind of calculator you are using, and I think that 
may be the problem.

From my research, it appears that the TI-83 and others have a true  
numeric derivative function, which gives the derivative as a number 
for a specific value of x using a difference quotient for a very small 
difference.  You would enter

  nDeriv( sin(x), x, x )

and get the value of the derivative of sin(x) with respect to x at any 
specific value x, which it can graph for you by repeating this 
calculation for each x.  The second argument to the function is the  
variable with which to differentiate, while the third is the value at 
which to evaluate an approximate derivative.  This is done by taking 
an approximate limit

                              f(a+h) - f(a-h)
  nDeriv( f(x), x, a ) = lim  ---------------
                         h->0       2h

You got an expression instead, which suggests that you are using a 
TI-92 or equivalent.  That actually produces a difference quotient 
expression, rather than just a value, and has a slightly different 
syntax.  You probably entered the same thing you would enter on a 
TI-83 to graph the derivative,

  nDeriv( sin(x), x, x )

and got the expression

  sin(2x)
  -------
    2x

This is because

  nDeriv( f(x), x, h )

returns the expression

  f(x+h) - f(x-h)
  ---------------
        2h

which is the symmetrical difference quotient.  Since you unwittingly 
entered x for the difference h, this would become

  sin(x+x) - sin(x-x)   sin(2x) - sin(0)   sin(2x)
  ------------------- = ---------------- = -------
          2x                   2x            2x

This is not the derivative, but a specific difference quotient with a 
large value of h.  You need to make h small to get an approximation to 
the derivative.

Isn't it nice of TI to make two calculators with the same named 
function that does two entirely different things?

To do what you want on the TI-92, drop the third argument, so that it 
will default to h=0.001, and just graph the resulting function, which 
will be a good approximation to the derivative.

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/