00:01
Definition of the derivative to find the limit, to find the derivative here.
00:05
So that means we want to take the limit as h approaches 0 of f of x plus h minus f of x, all that over h.
00:12
So what we're going to do is do some substitution.
00:15
The first thing i'm going to do, let's take our limit is h approach is zero.
00:19
We're going to put x plus h in place of x here.
00:22
So that's going to be f.
00:25
Let's go back up.
00:27
I've got my eraser in a different place here.
00:29
That's going to to be the square root, pin, pin, pin, there it is, square root of x plus h squared plus one.
00:41
Now we're going to come over and subtract x squared plus one and we're going to put that over h.
00:48
Now we have a problem.
00:49
We cannot put zero in place of h right now because it won't, it be undefined.
00:56
So what we have to do is rationalize, but this time we're going to rationalize, we're used to rationalizing the denominator, now we're going to rationalize the numerator.
01:05
So be x plus h squared plus 1 plus the square root of x squared plus 1.
01:11
So i'm going to put that here as well.
01:16
It looks horrible, but it's really not so bad.
01:20
So now in the numerator we have that difference of two squares.
01:23
So i'm going to take this piece, who thought i was getting my pen, we're going to take this piece and this piece, multiply it together.
01:31
So it's going to basically eliminate the square root...