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Derivative of Tan x

Date: 03/31/2001 at 23:50:29
From: Tong
Subject: Derivative of tan x

I know that finding derivatives can be found in two ways. One is by 
taking the limit of 

  lim h>0   -------------

Can you show me how to get the derivative of tan x by:

            tan(x+h)-tan x
  lim h>0   --------------

I tried many ways but still can't work it out.

Date: 04/01/2001 at 12:16:29
From: Doctor Jaffee
Subject: Re: Derivative of tan x

Hi Tong,

First of all, the easy way to find the derivative of tan x is to 
calculate the derivative of sin x/cos x using the quotient rule of 
derivatives. It works out quickly and easily. However, that is not 
what you asked, so here is my suggestion for proving that the 
derivative is the sec^2(x) using the definition of derivative that you 
are using.

As a preliminary step, recall that       sin h
                                  lim    ------ = 1 
                                  h->0     h
Also, cos 0 = 1

                            tan x + tan h 
Furthermore, tan(x + h) =   --------------
                            1 - tan x tan h

So, try again. Use the formula for tan(x + h) that  I have above, 
transform tan h to sin h/cos h, and use the limit of sin h/h that I 
mentioned above.

The result should be that everything cancels out except 1 + tan^2(x), 
which equals sec^2(x).

I hope you find this helpful.  If you get stuck, write back and show 
me everything you have done so far and I'll try to get you back on 

- Doctor Jaffee, The Math Forum   
Associated Topics:
High School Calculus

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