00:01
Okay, so for this question, we're asked to find the derivative of f of x using the formal definition of derivative, and we're given that our f of x is going to be equal to x squared plus x.
00:13
So the important part here is to remember what the formal definition of derivative says.
00:19
So the formal definition of derivative says f prime of x is going to be equal to the limit as h approaches 0 of f of x plus h minus x.
00:30
F of x all over h.
00:33
So now all we have to do is figure out what is f of x plus h.
00:39
So if f of x is equal to x squared plus x, that's going to tell us that x plus h, that quantity squared plus x plus h, is going to be our f of x plus h.
00:53
Then our f of x is still the same as it was.
00:56
So now let's look at that limit.
01:00
So as the limit, so h approaches 0 of our f of x plus h, so x plus h squared plus x plus h minus the entire f of x, so minus the entire x squared plus x all over h.
01:22
So usually when we're doing limits, we want to just plug in what our variable is approaching, in this case it be zero.
01:30
However, that is kind of a no -go right now because we have, we would have zero in the denominator, which we cannot do.
01:38
So that tells me, let's go ahead and expand out this entire top and see if anything magically cancels.
01:46
So that would be 2x squared plus 2xh plus h squared plus x plus h minus x squared minus x because this negative distributes...