What is a Derivative?

Date: 06/07/98 at 16:49:52
From: Max
Subject: Definition of a derivative

What is the concise definition of a derivative? I kind of know the 
meaning, but I can't see it visually. Do you have any ideas that 
will help?

Date: 06/07/98 at 17:51:45
From: Doctor Pat
Subject: Re: Definition of a derivative

Max,

I'm not sure if I completely understand the question, so I will answer 
both sides of what you might have meant.  

First, you need to think of a derivative as measuring the rate at 
which something changes as measured by something else. We do that a 
lot in life and never think of it as a derivative. For example, in a 
car, the speedometer is a measure of the change in position per unit 
of time. In this case we would write dP/dt where P stands for position 
along some path of travel (in miles) and t stands for time (in hours).  
Of course the speedometer only works correctly when you are going 
forward. In airplanes they have a gauge that measures the same thing 
for elevation. It measures how fast you are going up (or down), which 
is important as you can imagine. 

When you see the stock market report on the TV in the evening, it 
measures the change in the aggregate stock index per unit of time. In 
this case the units are dollars per day.  

On a graph, the visual pattern is one of slope. If we measure the 
closing stock price from day to day we notice that the graph gets 
higher on days when the price change is positive, and lower when the 
stocks go down. The steeper the slope, the faster the change, and 
positive means I'm making money.  

This makes sense in terms of how the derivative is defined. The basic 
part of the formula for the derivative is just the formula for slope.  
The instantaneous part is where the limit notation comes in. Let's 
look at something simple like y = x^2.

If we wanted to find the derivative at x = 3, we could look first at 
the graph for a clue. Is the curve going up or down? Imagine a tangent 
to the curve at x = 3. The slope of the tangent line is the slope of 
the curve at that point, by definition.  

We find it numerically by taking two points, x1 and x2, and the 
associated y values y1 and y2. In our case x1 = 3 and y1 = 9. If we 
pick x2 to the right of x1, then the slope between the two points 
would be given by:
       
       y2 - 9
   m = ------
       x2 - 3  
  
To find the "instantaneous" change, we just let x2 get closer and 
closer to 3. This is where the limit part comes in: as x2 goes to 3, 
the change in x (which we call dx) goes toward zero.  

I hope I've been able to help. If not, drop us another note.

Keep trying,

-Doctor Pat,  The Math Forum
Check out our web site! http://mathforum.org/dr.math/