Find the derivative of tan (x) by definition

Question

Find the derivative of  f(x) =tan (x) by definition? 

Answer 1

If you need to do this by the definition involving limits, then this would be one way to go about it:

 

The derivative by limits definition is:

 

lim        f(x+h)  - f(x)

h -->0   ---------------

                    h

 

So

 

f(x) = tan x = sin x

                    --------

                     cos x

 

and

 

f(x+h) = tan (x+h) = sin(x+h)

                               ----------

                                cos(x+h)

 

Putting these into the definition:

 

lim         sin(x+h)        -      sin x

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

             cos (x+h)              cos x

         -----------------------------------

                              h

 

Need a common denominator for top subtraction, so multiply left fraction by (cosx/cosx)  and multiply right fraction by cos(x+h)/cos(x+h).  This results in

 

lim          sin(x+h) cos(x)     -    sin(x) cos(x+h)

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

                        cos(x+h) cos(x)

       ------------------------------------------------------

                                     h

 

Now "fix" the complex fraction by multiplying the top fraction by  [cos(x+h) cos(x)]  to get rid of the denom in the top, and multiply the "h" in the bottom by  [cos(x+h) cos(x)] so that you have really just multiplied the whole fraction by 1.

 

 

lim             sin(x+h) cos(x) - sin(x) cos(x+h)

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

                          h cos(x+h) cos(x)

 

 

Right here, we need a trig identity.  

 

sin(A-B) = sin(A)cos(B) - cos(A)sin(B)

 

This is our situation if A = x+h  and B = x, so can rewrite the numerator as sin((x+h) - x) and this is sin(h)

 

Putting it all together:

 

lim           sin(h)

h-->0     ---------------------------

               h cos(x+h) cos(x)

 

Grouping this very carefully, we can use the fact that

 

lim             sin(h)

h-->0       ----------    = 1

                    h   

 

So

 

lim       sin(h)       1              1

h-->0 --------- . ------      .  --------

              h       cos(x+h)    cos(x)

 

=            1  .       1         .        1

                     ---------         --------

                       cos(x)          cos(x)

 

= 1 . sec(x)  . sec(x)  =  sec²x

 

Answer 2

Tan = sin/cos.  Use the quotient rule to find the derivative

 

[(cos)(cos) - (-sin)(sin)]/cos²

 

(sin² + cos²)/cos² = 1/cos² = sec²