 Hello, and welcome to a screencast about finding the slope of a tangent line using implicit differentiation. Okay, implicit differentiation, that's a mouthful for sure. So what this is really saying is that you've got y in your function, like down here, but y is really some unknown differentiable function of x. Okay, so remembering in the last screencast where you saw us like f of x written, and then you knew that the derivative of that was simply f prime of x. Even though you didn't know what f of x was, you knew that it had a derivative and you were going to call that f prime. Okay, we're going to be doing the same thing with y then, so y is going to be considered like your outside function. And then whenever you do the derivative of that, you're going to be calling that dy dx. Okay, and dur here stands for derivative. Okay, so again really with this stuff, you're thinking about the chain rule. And that I think is the hardest thing to get through your mind whenever you're looking at these functions. Okay, so just keep thinking chain rule, y is some unknown function of x. I don't know what it is, but I know it's got a derivative. And if you can get that through your head, then hopefully this will make it a little bit easier. Okay, the first example I want to look at today says for a curve given implicitly by x squared y minus 4x plus y cubed equals 1. I want to find the slope of the tangent line at the point 4, 1. Okay, so first of all, why is this function defined implicitly? Well, can you solve it for y? I don't think you can do it very easily. It's probably doable, but just not very nice. So because you can move this 4x over, sure thing, that would be no problem. But then you've got a y cubed and a y term in here. So trust me, this is going to be a big mess and you don't want to go there. Okay, so instead we're going to go ahead and do implicit differentiation. So what that says is we want to do the derivative of both sides with respect to x. So while I'm writing all of this out, I want you to be thinking about how can I do the derivative of these pieces with respect to x? Because that's going to be our next thing here. Okay, and that doesn't mind a sign. Okay, so x squared y, our first piece here, x squared y, what does that really mean? Well, this really means x squared times y. So in here I'm going to have to use the product rule for the first piece. Okay, I've got a minus and I've also got a plus here, so we're just going to apply the sum rules, so that's no problem. 4x, I can take the derivative of that piece, no problem. Y cubed, we can take the derivative of that piece, no problem, assuming we remember that we've got y as some unknown function of x. Okay, so let's go through and do the derivative of each piece. So again, we want to do the derivative of the first piece, we said that was going to be a product rule. So I'm going to go ahead and write out what that's going to look like. So the derivative with respect to x of x squared, because you do the derivative of your first piece times your second, plus then we leave the first piece alone, the derivative with respect to x of our second piece. Okay, so these two pieces here came from doing the product rule of this first chunk. Alright, minus, then we need to do the derivative with respect to x of 4x. And then plus, we need to do the derivative with respect to x of y cubed. And then just for the heck of it, I'll rewrite the last part here, the derivative with respect to x of 1. Just so I'm consistent with where I do derivatives and where I rewrite pieces. Okay, so now let's start doing derivatives. The derivative with respect to x of x squared, 2x. And then times our y piece here from the product rule. Plus, okay, x squared, no problem. Derivative with respect to x of y. Okay, so think about y as your outside function. So what happens when you do the derivative of y? It becomes 1. Times, now we need the derivative of y with respect to x. Well, like I said up here, that's just going to be written as dy dx. Minus, now the derivative with respect to x of 4x, that's just 4. Plus, derivative with respect to x of y cubed. Okay, well y cubed is like our outside function. So if we do the derivative of that, so that's going to give us 3y squared. Now we have to multiply by what's the derivative of y with respect to x? Well, dy dx. So that's really the derivative of our inside function, because we don't know what that function is. Equals, then the derivative of 1 is simply 0. Okay, so I think probably getting to this stop is going to be the hardest for some of you until you practice. And again, until you keep the idea in your head that you're using the chain rule in here. Y is some unknown function of x, so you've got to make sure to account for that when you go to do your derivatives. Okay, but you notice for this one, I don't have like a y prime equals or a dy dx equals or anything like that. But we're trying to find the slope, and that's really what we're trying to find here, right? Slope means derivative of y with respect to x. Well, I got to solve for it now. So you notice this piece doesn't have anything in it. This piece has a dy dx, this piece doesn't, and this piece does. So now we just need to do a little bit of algebra, and we're going to have to solve for dy dx. So I'm going to move the first piece and the third piece over to the right side, and then I'm going to factor a dy dx out of the two pieces here that are left. And if this is too much algebra for you, I'll go ahead and do it in two steps. So I'm going to move this piece over, so I'm going to subtract, let me do this in a different color. I'm going to subtract a 2xy from each side. Again, to kind of keep things balanced. And then I'm going to add a 4 to both sides, again, to kind of keep things balanced. Okay, so that'll wipe out here and here. So now the two pieces that I have underlined in green here have a dy dx in them. So let's go ahead and factor that part out. And that leaves me with x squared times 1. Okay, well, that's just x squared plus. And then here we've got 3y squared times this. So that'll be a 3y squared. And then equals negative 2xy plus 4, since 0 doesn't change anything. Okay, remember we're solving for dy dx. So now we're going to have to divide both pieces, or both sides by that piece. And I think we're almost there. So negative 2xy plus 4 all over x squared plus 3y squared. Okay, so that is our derivative of y with respect to x. But let's go back and read the problem again. My highlighted slope, it wants the slope of our tangent line at 4, 1. So that means I'm going to have to go down here now and to figure out my slope, I'm going to have to actually plug in those values. So negative 2 times 4 times 1 plus 4 all over 4 squared plus 3 times 1 squared. Okay, so checking out the algebra here. Let's see, I believe that gives us a negative 4 in the numerator and a 19 in my denominator. So that number there is the slope of my tangent line to this crazy function at the point 4, 1. Thank you for watching.