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### 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/
```
Associated Topics:
High School Calculators, Computers
High School Calculus

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