Finding First Derivatives by Definition

Date: 10/30/98 at 22:13:53
From: Lisa
Subject: Finding Derivatives by Definition

I can't seem to understand how to find the first derivative by 
definition. The part of the equation I can't figure what f is in 
f(x-h) - f(x).

Could you please show me a good sample problem? I can do it the easy 
way, but our teacher says we have to know how to do it by definition 
on our tests.

Date: 10/31/98 at 03:19:34
From: Doctor Pat
Subject: Re: Finding Derivatives by Definition

Lisa,

Here is a step-by-step approach that makes it manageable (NOTHING 
makes it easy.) I'll use the very easy f(x) = x^2 + 2x for my example.

1) Write out f(x) and f(x+h) to the side and simplify both:

     f(x) = x^2 + 2x   
   f(x+h) = (x+h)^2 + 2(x+h) = x^2 + 2xh + h^2 + 2x + 2h  
       Note (x+h) replaces x everywhere it appeared.

2) Now write out the Newtonian quotient with f(x+h) and f(x) as above:

   f(x+h) - f(x)   [x^2+2xh+h^2+2x+2h] -[x^2+2x]
   ------------- = -----------------------------
         h                      h

3) Simplify numerator as much as possible. Note that you should try to 
   find an h in every term:

   2xh + h^2 + 2h     
   --------------
         h

4) Now divide h in the numerator and denominator:

    2x + h + 2
    ----------
        1

5) Finally, evaluate the limit as h goes to zero:

   Lim  2x + h + 2 = 2x + 0 + 2 = 2x + 2  
   h->0

Sometimes the simplification in steps two and three takes much more 
work, but this is a useful pattern to work from.

Good luck.

- Doctor Pat, The Math Forum
  http://mathforum.org/dr.math/