 What I want to do in this video is something similar to a previous video We did where I'm going to give you the graph of function f. We're not going to give you a formula We're just looking at the graph right here, and we want to identify What does the derivative of this graph f look like based upon the graph f? So the first thing we're going to do is we're going to look for where places where the derivative would be undefined So we've learned previously that the derivative so f prime of x here It does not exist when a couple of things happen first you have discontinuities Right or if there's any place where the function is discontinuous where f is discontinuous then the derivative will likewise be Undefined for which we look at this graph right here. It's completely continuous. There's no problem with continuity whatsoever Another thing we have to look for is going to be sharp corners Places where the function is not smooth and we see a couple of those And we see at least one here on the graph you can see there's a sharp corner right here Another place which might be suspicious would be right here, but you actually see that the transition from Curve to flat is actually smooth. There's no corner right there So we did get one place where the functions discontinued or what the derivative will be undefined And so it's not differentable at that point x equals 3 and then the other thing we have to look for is going to be vertical tangent lines Vertical tangents which our graph has we can see some of those right there There actually would be a vertical tangent line right here, but it's already a corner So we're throwing that one out. There's gonna be a vertical tangent line right here at x equals negative two There's also a vertical tangent line at x equals negative six So if we summarize What we've discovered so far we see that our function will be the derivative will be undefined at x equals negative six at x equals negative two and at three and So some things we should then compare when we look at these graphs together Is that when you have a vertical tangent line on the function that corresponds to a vertical acin tote to the derivative? So when you take x equals negative six, you're gonna get a vertical tangent line at the location So I'm just gonna add that to my picture here The picture on my right is gonna be the derivative so at x equals six There's a vertical tangent line. There was also one at x equals negative two negative two And then also like I said at this point at this point over here three if you approach from the right There is gonna be this vertical tangent line There's also a horizontal tangent line from the left again This is the reason why sharp corners are a problem the left approach of the tangent line will disagree from the right approach The left approach wants to be horizontal the right approach wants to be vertical So we're gonna see something very bizarre in our graph when we look at that But it turns out there's gonna be a vertical acin tote on the derivative likewise at x equals three So we're gonna prepare for those things as well So let's look at some other weird behavior that's happening on this graph What's happening here at x equals three because of the sharp corner the approach from the left wants to be zero The approach from the right wants to be Vertical acin tote so what we see is as you approach x equals three from the left Well everywhere in this sector right here is gonna be flat. It's completely flat It's a constant region the derivative is gonna be zero in all of those locations So what we see on our graph as we try to plot it is that from x equals Zero to x equals three the function is completely flat We see this right here completely flat And I'm gonna put an open dot right there because the function is undefined at x equals three on the other hand if you're to the right of F of x are right of excuse me x equals three. We're gonna have this vertical tangent line It's gonna be important whether it's positive or negative or a little bit to this to the to the right there We see that it's gonna be a negative slope and so what this tells us is so we approach acin tote three on the derivative from the right We're gonna be going towards negative infinity So be coming down like this and then if we kind of finish this up What's happening as we can closer and closer to x equals six it's gonna get flatter flatter flatter There's gonna be a horizontal tangent line at x equals six and so that's gonna be an x intercept on our graph right here So we see with our function. I'm gonna erase this part and try again We're gonna see some type of asymptotic behavior looking something like this. All right come back to this point right here X equals zero is kind of curious to us and also negative two is curious We see that when we're a little bit to the left of negative two Right, we're gonna have a positive slope That's a positive tangent line very very steep So we're gonna be really close to infinity over here But then as we get closer and closer to zero this thing's gonna flatten out to be a horizontal So what we see would be a function looks something like this So these this will come here and touch at the origin zero zero and as you approach x equals negative two from the right You're gonna be positive infinity. All right, so the last thing I want to show you here is notice that at negative four There's a horizontal Tantent line that's gonna be an x intercept of the derivative So we anticipate an x intercept right here if you're a little bit to the right Excuse me if you're a little bit to the left of negative two that's gonna be a positive Tangent and so we should be close to infinity over here on the other hand If you are a little bit to the right of negative six, that's gonna be a negative slope And so we should be a little bit over here So kind of putting those together and connecting in a nice smooth way We're gonna get a picture that kind of looks like a tangent curve We get something like this here in green and this sort of monstrosity of a function Would be the graph of f prime right here I don't want you to necessarily take my word for it and also because my picture might not be the best drawn right here Let's look at a computer generated image of these things So you see again the function f that we had from before and now graph to the right We have a picture of its derivative with all those behaviors We saw there and so this can get a little bit messy a little bit confusing But the important part is that we can derive the picture of the derivative from the function f And that's why we call it the derivative because it is derived from that original function And so that's gonna end our trilogy in lectures 16 17 and 18 I hope you enjoyed all of this learning about the derivative for our lecture series math 12 10 and chapter 3 We're gonna talk a lot more about the derivative and focus on techniques We can use to make our computations of derivative much more fetch much more efficient So, please take a look for those links right now if you've learned anything about the derivative in these videos feel free to hit the like button and Subscribe if you want to see more videos like this in the future as always if you have any questions Please feel free to post them in the comments and I'll be happy to answer them as soon as I can have a great day everyone. Bye