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### Derivatives

```
Date: 10/28/96 at 12:53:53
From: Kandice Longhurst
Subject: Need help with Calculus!

Hello,

I am taking calculus this year and I don't understand the concept of
the derivative.  Could you explain it to me?  It would be very

Thank you.
```

```
Date: 10/29/96 at 1:3:21
From: Doctor Daniel
Subject: Re: Need help with Calculus!

Hi Kandice,

The concept of a derivative is really very important and I hope I can
help explain it to you.  Unfortunately, mathematicians use it so often
that we sometimes assume that this idea, like many other important
ones in math, is so obvious that it should be immediately clear.  But
it's really not.

Let's say you were given a function f, which is a function from real
numbers to real numbers. (I'm hoping you're pretty clear on the idea
too; it's another extremely important idea which people don't always
get.) Anyhow, we have this function f : R -> R. (That notation is
just shorthand for f maps from reals to reals.) Maybe, to put a
concrete feel to it, f is a function that maps the time, since an
experiment began with the amount of radioactive material in a source
which is decaying.  But the idea is general enough to just work for
any function from R -> R.

Now, suppose you want to find how fast the function is changing at any
given moment. For example, you might be curious how fast the source is
emitting radioactive particles. This would be measured in particles/
sec, and it should be fairly easy to compute, all things considered.
How would we try doing it?

Well, let's say we wanted to find the rate of change of the function
at the time = 10 seconds. We could compute the amount of particles
left in the sample after 10 seconds, f(10), and the amount of
particles left after 20 seconds, f(20).

f(20) - f(10) is the amount of particles emitted in 10 seconds.
If f(x) is a straight line, then the rate of change at x = 10 seconds
will be roughly:

f(20)-f(10)
-----------
20-10

This is the slope of the line from (20, f(20)) to (10, f(10)). But
suppose it's not a line. Then the rate of change at 10 sec is probably
closer to f(11)-f(10), which is the slope of the line from (11, f(11))
to (10, f(10)).  In fact, if we get closer and closer to 10, we're
computing the slope of a line that is closer and closer to what the
function looks like closer and closer to when x = 10.  So, for
example, 10 * (f(10.1)-f(10)) is closer to our desired rate of change
than 1/10 * (f(20)-f(10)).

Remember that we're really just computing the slope of lines that look
a lot like our curve.  What we want is the slope of the curve itself.

If you're pretty clear on the definition of a limit, you should see
the answer at this point.  The slope of the curve ("the rate of change
of the function f","the derivative of f") at x = 10 will be this
limit:

lim     (f(10+y)-f(10))
y -> 0  ----------------
y

The denominator and numerator are both approaching zero!  In practice,
however, the value of the numerator will include something with a
power of y which will cancel the y on the denominator, and we'll be
fine.

This is the definition of the derivative of f in general, rather than
just at x = 10:

f'(x) =  lim     (f(x+y)-f(x))
y -> 0  ----------------
y

What do we know that this must mean?  Well, for starters, f has to be
a continuous function at the point we're considering. Do you remember
what continuity means?  That means you could draw the graph of the
function without ever lifting a pencil off the paper. In language of
limits, that means

lim   f(x+y) = f(y)
y->0

Otherwise, the numerator won't be zero and the limit will be infinite.
Another requirement is that the function doesn't change directions
suddenly as you get closer to x.  Another (though it's obvious) is
that the function has to have a value at x; that is, f(x) has to
exist.

If the derivative has a value at all points, then it's clear that it's
also a function. So, for our function f, we can talk about the
function f', which is its derivative, where f'(x) is the derivative of
f at the given point.  You've probably learned formulas by now which

That's a very long answer to a very important question.  The
derivative of the function at a given point is the slope of the curve
at that point. The way we compute it is by computing the slope of line
segments which get closer and closer to the point itself, and
eventually taking the limit of this process.  It's a function itself,
and in practice, rather than computing the limit directly, we compute
it using formulas which make our lives much easier.

I hope this helps.  If you're still confused, please don't hesitate to
and try out the kind of construction I went through with words above.

Good luck, Kandice!  Happy math!

-Doctor Daniel,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Calculus
High School Definitions

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