Meaning of Derivative

Date: 10/12/96 at 18:6:19
From: Dominic Tsang
Subject: Meaning of Derivative

Can you give me a plain English meaning of the idea of a derivative? 

Thanks!

Date: 10/13/96 at 15:54:4
From: Doctor Scott
Subject: Re: Meaning of Derivative

Dominic,

The derivative does, in fact, have a nice "plain English" meaning. 
You probably learned that the derivative is the slope of the tangent 
line to the curve at a point.  The derivative describes the rate of 
change of the function at that point.  Actually, it is the 
*instantaneous* rate of change of the function at the point.  

For example, if you are blowing up a balloon, the volume of the 
balloon depends on the radius of the balloon.  That is, 
V = (4/3)*pi*r^3.  The derivative of V (with respect to r) would 
tell you how fast the volume is changing as the radius changes.

Hope this helps!

-Doctor Scott,  The Math Forum
 Check out our web site!  

Date: 10/13/96 at 16:2:48
From: Doctor Ken
Subject: Re: Derivative

Hi Dominic -

Here's an example of the derivative and what it means.  Let's say 
you've got a function, call it f(x).  The derivative of f, f'(x), 
tells you how fast f is changing.  If the derivative is positive, 
f is increasing, and if the derivative is negative, f is decreasing.  
If the derivative is 0, then at that point f is neither increasing 
nor decreasing.

Let's say f(x) represents the position of a car on a straight road at 
time x.  Then the derivative f'(x) tells you the velocity (that's 
like speed) of the car at time x.  If the car is going forward the 
velocity will be positive, and if it's going backward the velocity 
will be negative.

If you take another derivative, you'll call it f''(x), and that tells 
you the acceleration of the car at time x.  If the car is speeding up 
the acceleration will be positive, and if it's slowing down it will be 
negative.

Here's a little-known fact I'll let you in on: if you take another 
derivative, that has a name too.  It's called "jerk," and we write 
f'''(x). Can you figure out what the physical interpretation of jerk 
is?

The derivative is used all the time in physics, in exactly this kind 
of way.

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