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```Date: 05/13/2003 at 14:24:38
From: Kim
Subject: Derivative as a function

The problem asks us to sketch graphs of f and f' for f(x) = sin x and
to guess the formula for f'(x). Could you explain how you would work
this problem out?

Thank you.
```

```
Date: 05/14/2003 at 13:46:37
From: Doctor Dotty
Subject: Re: Derivative as a function

Hi Kim,

Thanks for the question,

The derivative of a function at a point is the gradient of the curve
at that point.

If f(x) = Cos x, then the graph of f(x) is:

|
..1..
.'  |  '.
.'    |    '.
-3Pi/2   -Pi    -Pi/2.'      |      '. Pi/2    Pi    3Pi/2
'.''''''''''''''''':''''''''|'''''''''.''''''''''''''''.
'.             .'         |         '.              :
'.          .'           |           '.          .'
'..    ..'             |             '..    ..'
''''               -1                ''''
|

So what is the graph of f'(x)? We look at the gradient.

Well, at -3Pi/2, it is negative but increasing. It goes through zero
at -Pi, but starts decreasing again at -Pi/2. It goes through zero at
zero, but starts increasing again at Pi/2. It goes through zero at Pi,
but starts decreasing again at 3Pi/2, and so forth.

So it looks like this:

|
|
.......     1                        ...
.''       '.   |                     .''
-3Pi/2   -Pi.'    -Pi/2   ''.|      Pi/2       Pi.'   3Pi/2
''''''''.'''''''''''''''''':'''''''''''''''''.'''''''''
.'                   |'..            .'
..'                     |   '.       ..'
'''                       -1     '''''''
|
|
|

This is a reflection of the graph of y = sin x about the y-axis; so at
a guess, I would say that f'(x) = -sin x.

Can you sketch the derivative of sin x in a similar manner?

Write back if I can be of any more help on this or anything else.

- Doctor Dotty, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Calculus
High School Trigonometry

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