 Hello and welcome to a screencast about the limit definition of the second derivative. So this one's going to be very similar to what we did with the first derivative video, except we're going to have to use the limit definition twice. So the question of the day is, how do you compute the second derivative of a function? Again, using this limit definition. And don't worry, there is a light at the end of the tunnel, I promise. You'll be soon learning how to do derivatives without this limit definition. But for now we need to really make sure that we understand it. So as we've given before, the first derivative is defined to be f prime of x is equal to the limit as h approaches 0 f of x plus h minus f of x all over h. The second derivative is defined in a similar fashion. So the second derivative of x is equal to the limit as h approaches 0, the first derivative at x plus h minus the first derivative all over h. So if we're given a function x to the fourth plus 3x squared minus pi, I want to find the first derivative, obviously first, of this function before we can find the second derivative. So it's kind of like, yeah, you got to put your socks on before you put your shoes on. Okay, so f prime of x is equal to the limit h approaches 0. So by now you should be old pros at figuring out this limit definition. So we have to plug x plus h into the two pieces of our function where I see an x. So that'll be x plus h to the fourth power plus 3 times x plus h squared minus pi. Now I'm going to have to subtract, and again because my function comes in more than one piece, I'm going to throw in a parenthesis x to the fourth. So it's a 4 plus 3x squared minus pi all over h. Okay, so again comes some algebra fun. So I've got my limit h approaches 0. I need to do x plus h to the fourth power. That means I'm going to have to multiply out x plus h times itself four times. Well, if you watched our previous video, we did it as a cubic. So then you can go ahead and take that polynomial that we got and just multiply that by x plus h. Or you can, you know, just multiply it all out on your own. So anyway, when you do that, you end up with x to the fourth plus 4hx to the third plus 6x squared h squared plus 4h cubed x plus h to the fourth. I hope I don't run out of room. Okay, now we're going to have to multiply out the x plus h squared. So that's x squared plus 2xh plus h squared minus pi. And then I want to distribute my negative sign through as well. So that's going to end up with minus x to the fourth minus 3x squared. Cooperate, please. And then plus pi. And all of this, I barely had enough room for it all. Woo, that's not straight, but that's okay. All right, it's all over h. Okay, so now we have to do some algebra here. I think I'm going to go ahead and start cancelling some things only because this is so ginormous. So I'm going to use a different color here. So my x to the fourth are going to cancel and my pi's are going to cancel. And unfortunately for now, that's about it. Okay, let me start back here this time. All right, we have to limit h approaches zero. So I have 4hx cubed plus 6x squared h squared plus 4h cubed x plus h to the fourth. And again, at any point, it might be a good idea just to pause the video and see kind of where your algebra skills are at as well on this. Then you can come back and watch me and make sure that we agree. So distribute my 3 through, that'll give me a 3x squared plus a 6xh plus a 3h squared and then minus 3x squared. Then all of this craziness is divided by h. Okay, I think I see one more piece I can cancel and that's my 3x squared. And just to save ourselves a little bit of work here. And again, hopefully you've had the practice with these so you can understand what I'm doing. But again, you should notice that all, let's see, 1, 2, 3, 4, 5, 6 terms that we have left all have an h in them. So let's factor that out. So that leaves us with 4x cubed plus 6x squared h plus 4h squared x plus h to the third plus 6x plus 3h. All right. And then all of this is divided by h. Again, we can go ahead and cancel those h's. And then what we have left in these parentheses is now a nice polynomial. We don't have anything funky going on. So if we let the limit as h approaches 0, then our derivative ends up being 4x cubed plus 6x. Because all the rest of the terms have an h in them, so those terms are going to be going to 0. So that gives us our derivative. Okay. Now we get to do it again. So let's go down here. I've got more room for us. So our second derivative is going to be defined as the limit h approaches 0. So now we have to take our function up here that we just found, our first derivative function. And we've got to plug x plus h into that. So that'll be 4 quantity x plus h cubed plus 6 quantity x plus h, and then minus that function. So that'll be 4x cubed plus 6x, and then all of this is divided by h. Okay. So now we've got to do some algebra with this stuff. And I'm going to start over here this time. Limit h approaches 0. So 4 times, remember x plus h quantity cubed means you have to multiply that out 3 times. So when you do that and collect all your like terms, you'll end up with x cubed plus 3x squared h plus 3x h squared plus h cubed. Distribute my 6 through, so that'll give me 6x plus 6h. Distribute my negative through minus 4x cubed minus 6x. And then all of this is divided by h. Okay. Almost there. Let's see. I think I will go ahead and cancel my 6x's just so I don't have to write those twice. Let's go ahead and distribute my 4 through. So limit h approaches 0, 4x cubed plus, that gives us 12x squared h plus 12x h squared plus 4h cubed. Okay. Then I've got my plus 6h minus my 4x cubed. And all of this is over h. And again, we can cancel our 4x cubes out. And we have 1, 2, 3, 4 terms left. They all have an h in them. So let's go ahead and factor that h out. So that leaves me with a 12x squared plus 12x h plus 4h squared plus 6, so I lost my 4. And all of this is divided by h. Again, I can cancel my h's out. And we have a nice polynomial left in our parentheses. We can let h go to 0, so that's going to wipe out my two middle terms. So my second derivative then is 12x squared plus 6. We did it. All right. Thank you for watching.