Expert: Paul Klarreich - 2/11/2008

Question
My question asks:

Using the definition of the derivative, i.e. (f(x+h)-f(x)]/h and taking the limit as h -> 0 for the function f(x)= cos (x)?

There is a hint that I will need to use the angle sum formula cos(A+B) for this problem to do this. But, I have no idea what that means and my text does not help me out with this either.  

Answer
Questioner:   Andrew
Category:  Calculus
Private:  No
 
Subject:  Trigonometric Derivative..
Question:  My question asks:

Using the definition of the derivative, i.e. (f(x+h)-f(x)]/h and taking the limit as h -> 0 for the function f(x)= cos (x)?

There is a hint that I will need to use the angle sum formula cos(A+B) for this problem to do this. But, I have no idea what that means and my text does not help me out with this either
.....................................
Hi, Andrew,

Actually, the 'hint' suggests that formula -- there are other formulas.  Did the book actually give the formula?  If not, there are plenty of places to get it:

cos(A + B) = cos A cos B - sin A sin B  [Sum Formula]

Do you know how to use a formula?  It is a PATTERN -- it contains place-holders wherein you put items from your actual problem, to form an INSTANCE.  In this case, when you start your derivative calculation:

f(x+h) - f(x)
------------- =
    h

you do just what those symbols tell you to do.  f(x+h) means take the PATTERN for f() and replace its one place-holder with the expression (x+h).  So:

cos(x+h) - cos(x)
------------------ =
      h

Now you do more pattern-matching.  cos(x + h) matches the left side of the sum formula.  So do the instantiation -- x matches A, h matches B.

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

BECOMES:

cos(x + h) = cos x cos h - sin x sin h  [Sum Formula]

and now you have your derivative. (Well, almost.)

cos x cos h - sin x sin h - cos(x)
---------------------------------- =
          h

You do a bit of simplification:

cos x(cos h - 1) - sin x sin h
---------------------------------- =
          h

cos x(cos h - 1)     sin x sin h
----------------- -  ----------- =
       h                h  

     (cos h - 1)           sin h
cos x ----------- -  sin x  ------ =
          h                  h  

Now you were supposed to have ALREADY LEARNED that:
     cos h - 1
lim   ---------- = 0, and
h->0      h

     sin h
lim   ----- = 1.
h->0    h

from which you get your derivative of  - sin x