Expert: Paul Klarreich - 11/23/2007

Question
I have a question about how to make the verfication of the derivative of tan(x)

with the lim  tan(x+h)-tan(x)
        h->0 ---------------
                     h
i know the answer is sec^2(x) but i don't know why.
Thank!! Antonio

Answer
Questioner:   Antonio
Category:  Calculus
Private:  No
 
Subject:  derivates-trigonometry
Question:  I have a question about how to make the verfication of the derivative of tan(x)

with the lim  tan(x+h)-tan(x)
       h->0 ---------------
                    h
i know the answer is sec^2(x) but i don't know why.
Thank!! Antonio
...................................
Hi, Antonio,
 
Here is one way to do it:

lim   tan(x+h)-tan(x)
h->0 --------------- IS D(tan x); that's what the notation means.
          h

Use these facts:
       sin x
tan x = -----
       cos x
         1
sec x = -------
       cos x

D(sin x) = cos x
D(cos x) = - sin x

and use the quotient rule:

D(tan x) = D(sin x/cos x) =

(cos x)(cos x) - (sin x)(-sin x)
--------------------------------
            (cos x)^2


cos^2(x) + sin^2(x)
---------------------
    (cos x)^2

   1
----------
(cos x)^2


sec^2(x)

Now if the ground rules for the problem say you can't do it like this, let me know and I'll see what I can come up with.
-----------------------
I have, in fact, come up something.  Try this proof:
First, note that
   tan h          sin h     1
lim -------- = lim ------- ------- = 1 / 1 = 1
      h              h    cos h
Of course, lim means  lim(h->0), to save typing.

Now use the identity:
            tan x + tan h
tan(x + h) = ------------------
            1 - tan x tan h

and we work out the limit:

tan x + tan h
-------------    - tan x
1 - tan x tan h
---------------------------
        h

tan x + tan h - tan x + tan^2(x)tan h
-------------------------------------
h(1 - tan x tan h)

 tan h   + tan^2(x)tan h
---------------------------
h(1 - tan x tan h)

Use the identity: tan^2 = sec^2 - 1:


tan h   + (sec^2 x - 1)tan h
-------------------------------------
h(1 - tan x tan h)


tan h   + sec^2 x tan h - tan h
-------------------------------------
h(1 - tan x tan h)

+ sec^2 x tan h
-------------------
h(1 - tan x tan h)

Rearrange:

+ sec^2 x            tan h
-------------------  ------
(1 - tan x tan h)      h


Now lim tan h/h = 1 and  lim (1 - tan x tan h) = 1
and that does it.