00:01
The slope of a tangent line is this limit definition, which is really just the limit of a sequence slope.
00:09
So i've written this definition, and we're going to use this to find the tangent slope at various choices of x.
00:15
So in particular, the slope of our tangent line is the limit as h approaches zero of our function evaluated at c plus h.
00:26
So our function being x cubed minus 3x would be c plus h cubed, minus 3c plus h.
00:36
Now we're going to subtract our original function evaluated at just c.
00:40
Well, that would be c cubed minus 3c.
00:44
And we'll divide all of that by h.
00:49
Next up, we're going to expand.
00:52
So let's expand c plus h cubed.
00:56
Well, that's c plus h times c plus h times c plus h.
01:03
Let's see what that gets us.
01:06
Note i'm writing the limit every time, even though we haven't done that.
01:09
That's important to write in every step.
01:12
So if we expand c plus h by c plus h, we'll get c squared to ch and an h squared.
01:21
And that we need to multiply by yet another c plus h.
01:26
So i'm writing all the previous steps, even though we're sort of focused on the first part.
01:33
So we'll limit.
01:34
Okay, we're ready to fully expand.
01:36
So let's give every quantity a c first and then an h...