3-2-25
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Use the alternative formula for the derivative to find the
derivative of F of X equal three X / X - 1.
So the first thing we need to realize is the alternative
definition as the limit as Z approaches X of F of Z -, F of X
/ Z -, X.
So the limit as Z approaches X of three Z / Z - 1 minus three X
/ X - 1 all over Z -, X getting a common denominator three Z * X
- 1 - 3 X times Z - 1 all over Z - 1 * X - 1, and that whole
thing over Z -, X.
Now when we have that over Z -, X, we can really think of moving
that Z -, X into the denominator portion.
So if we distribute out, we get three ZX -3 Z -3 ZX plus 3X all
over the denominator.
The three ZX and the -3 ZX are going to cancel.
So we have -3 Z plus 3X over Z -, 1 X -1 Z minus X.
When I pull out a -3, we get Z -, X and the Z -, X on top and
bottom Cancel so -3 / Z - 1 * X - 1.
Now, the fact that we have Z approaching X means we're going
to put X in every time we see the Z so -3 / X - 1 * X - 1 or
-3 / X - 1 ^2.