00:01
In this problem, we're given the function f of x equals x minus 1 squared plus 1.
00:06
We're asked to find the derivative and then find the value of the derivative at the three points that are specified.
00:12
So to start by finding the derivative, we know that the derivative is given by this limit expression, the limit as h goes to 0, of f of x plus h minus f of x all over h.
00:27
So let's just plug in the function that we're given.
00:32
So capital f of x plus h is x plus h minus 1 squared plus 1 minus now f of x.
00:43
You just write down what f of x is, x minus 1 squared plus 1 all over h.
00:51
Now we're going to have to do a little bit of simplification here.
00:56
So this is the limit as h goes to 0.
00:58
Let's go ahead and expand everything in our numerator here.
01:06
So once we do this expanding, x plus h minus 1 all squared is x squared plus h squared plus h squared plus 2xh minus 2x minus 2h.
01:26
So if you expand that, then this is the expression you get.
01:29
And then there's another plus 1 all minus.
01:32
Now let's expand over here.
01:34
X squared minus 2x plus 1 plus 1, all over age.
01:43
This is pretty big, but we can just do some cancellation.
01:45
It's going to simplify very nicely...