Expert: Paul Klarreich - 10/3/2005

Question
if possible. can u e-mail me back asap
they deal with derivatives
1. find f'(prime)(a)...f(x)=(x^2+1)/(x-2)
2. find f'(a) square root of 3x+1
each limit represents the derivative of some function f at some number a. state such f and a in each case.

1. (tan x - 1) / (x-pi/4)
2. (t^4 + t - 2) / (t - 1)


Answer
Hi, Gina,

Your question lacks context, unfortunately.  I am not sure just what level of calculus you have reached.  I know it is first-semester, but:

1. Have you learned things like the product and quotient rule?
2. Are you required to use the 'limit process' or 'Definition of the derivative'?

Depending on your context, the approach to these problems is considerably different.

However, your second pair of questions suggests the second context, so that is what I will assume.
[VIEW IN A FIXED FONT.]

1.1: You want to write and evaluate:
           f(a + h) - f(a)
f'(a) = lim ---------------
      h->0       h

                      (a+h)^2 + 1
Now compute f(a + h) = -----------
                        a + h - 2
           a^2 + 1
and  f(a) = -------
            a - 2

Now carry out some algebra (which is too messy for me to type into the computer)

Remove parentheses in (a+h)^2
Subtract the fractions; the LCD is (a + h - 2)(a - 2)
(That's messy and tricky -- sorry about that)
Simplify as much as possible, then divide by h.
Finally, let h become zero.  You should NOT get a fraction with 0 on the top or the bottom.

1.2: f'(a), for sqrt(3x+1)

Carry our the same steps as above:

f(a + h) = sqrt(3(a + h) + 1)
f(a) = sqrt(3a + 1)

Write:  
           sqrt(3(a + h) + 1) - sqrt(3a + 1)
f'(a) = lim ---------------------------------
      h->0                 h

Simplify this.  The first step will be to RATIONALIZE THE NUMERATOR.  After that, things should simply (fairly) nicely.

======================================

There is an alternate form of the 'limit process':

           f(x) - f(a)
f'(a) = lim -----------
      x->a    x - a

which you could use in the examples above, too.
So how does each of these match up with the form you see?

For each one, look at the numerator and denominator.  Something on top should look like an f(x), while something else should look like f(some constant), and hopefully, that constant is on the bottom.


2.1 (tan x - 1) / (x-pi/4)
The tan x looks like a good candidate for f(x) and the pi/4 looks like a good candidate for 'a'.  Is it true that f(a) = tan(pi/4) = 1?  YES, IT IS.

So this one is the derivative of f(x) = tan x, at x = pi/4.
[Yes, that's all there is.]

After that, you can probably do this one yourself.
2. (t^4 + t - 2) / (t - 1)

Looks like f(t) = t^4 + 1, and a = 1.