Graphs and Gradients

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/