Differentiation

Date: 05/17/98 at 20:54:51
From: Ali
Subject: Trigonometry

I'm trying to find a GOOD definition for "differentiation."  I've 
looked in many math books, but none of them gives me what I'm looking 
for.  Please help!  

Thank you.

Date: 06/09/98 at 11:52:05
From: Doctor Jeremiah
Subject: Re: Trigonometry

Dear Ali:

My best answer to your question is:

   "Differentiation is the algebraic method
    of finding the instantaneous rate of change
    of a property.  This instantaneous rate of
    change is called the derivative."

Let me try to explain why:

If you have an equation that describes an object's property over a 
time period, the "derivative" is the rate of change of that property 
at any instant in the time period. If the equation is drawn on a 
graph, the derivative can also be determined by finding the slope of 
the graph at a particular point in time.

One example of this is when the property we are measuring is distance.  
Then the equation gives distance at certain times. If we differentiate 
the equation we get its derivative: the rate of change (distance/time) 
of the distance at a particular instant of time. The derivative is the 
velocity.

An example of using the graphing method of calculating the derivative 
is to graph the equation and find the slope at the point in time that 
you want the derivative.

For the distance graph, the slope is:

        distance at time 2 - distance at time 1
slope = ---------------------------------------
                    time 2 - time 1

The way to get a better derivative is to make the two times VERY close 
together.

But this gets really tiring when you need the derivative at many 
points on the graph. There has to be an easier way than graphing and 
calculating slope at every point we want. Remember I said there was an 
algebraic way of calculating the derivative (without graphing the 
equation)?  Well, there is: differentiation.

How to differentiate an equation is something you learn in calculus.  
If you want an example of using calculus to determine the derivative 
of an equation by differentiation it write me back.

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