Look So to get started with this. A big idea is just drawing a tangent line and estimating what that line looks like. So I drew the graph of technology. It's a Parabolas shifted up one unit On the graph is 2.102 which I've identified. So what does the tangent line look like? If I zoom in enough to this parabola at that .12, this little part of the graph will be extended and look something like this. So hopefully that tangent line makes sense to you that at that moment I have a function that's increasing. If I extend that little part of the graph to be a line, that's what the tangent line would look like. So what is the slope of the tangent line? Well, we can estimate it. It appears that another point on the tangent line graph is roughly two comma for So the slope is rise over run That's 4 -2/2 -1. to over one is to so roughly it appears that the slope of the tangent line is too. How could we think of slope a different way? Well, what we're encouraged to do is to just make a little table and see what happens when we consider The slope of a secret line between the .1, 2 and a point closer to that. So what if instead we considered The .1, 2 and then appoint At 1.01 And then 1.01 plugged in. Note, the function is X squared plus one. So if we plug in 1.1, We'd Square that and add one. So what's the slope among these two points? So what we're doing is picking two points super close together In considering the line between those two points. So the slope is the same will take the change in why? Over the change in X. So we get 1.01 squared plus one minus two. That's 1.1 squared minus one. Um That's practically zero, right? And one point oh one minus one Is also practically zero. 1.01 Squared Is very close to one. When we multiply that out, you get 1.0201. When we subtract one we get .0201 Over a .01. When we divide by .01, that's the same as multiplying by 100. So we get 2.01. So this second slope which just means the slope among two points. Is it essentially the tangent slope which we already estimated to be two? Lastly to finish this problem, let's consider a limit. So what is what are we really doing in the previous example? Let's consider the two points. 1, 2. And how about one plus a little amount. We'll call that little amount H and then that plugged into our function. So what we'd like to do is ask ourselves what happens when H goes to zero. So let's find the change in why? Over the change in X. So again each Justin notes Little run. How far away are we from that .12. So if we expand we get one plus two, H plus H squared minus one. Here are the ones cancel. So we get H. We notice these ones cancel. Leaving an H is a common factor in the numerator, which can cancel away with that ancient the denominator now is that H goes to zero, meaning it gets as small as we wish. The limit of this would be to which again is the canyon slope that we saw. So this calculation is essentially doing what we did above. But instead of 1.01, we have one plus any small amount. When that small amount goes to zero, we get to which is the tangent.