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### Differentiation

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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

Thank you.
```

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

Dear Ali:

"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/
```
Associated Topics:
High School Calculus
High School Definitions

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