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Meaning of Derivative

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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!
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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!
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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/
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Associated Topics:
High School Calculus
High School Physics/Chemistry

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