Hi and welcome to module for, we're now going to talk about what happens when you have two functions multiplied? How can you take the derivative more easily? We're going to start with simple functions that we can actually multiply out first and then we'll get to more complicated things later, but I just want to show you how it works. So if I have, I'm going to just going to call them f and g. So let's say f(x) is x squared minus two and let's say g(x) is x squared plus x. Okay, so if I have the function, let's say if I have h(x), which is f(x) times g(x). Right, so this is x squared minus two times x squared plus x, right? So in this case I could multiply this out if I wanted to find a cheap prime of x, I could first say, okay? Well, then h(x) is, let's see, I do the first one's x to the fourth, that one times that one plus x cubed minus two, x squared minus 2x. And then I could take the derivative, so we drop this power down to work by one same thing. And that was fairly simple because these were fairly simple functions to multiply out, but suppose let me give you a formula. So if I want to take the derivative, I'd better do it over here of f times g and I'll write it this way. The derivative of the product is going to be the first times the derivative of the second plus the derivative of the first times the second, okay, so let's see if that works. Okay, so in this case if I look at f, the derivative of f would be 2x and that would be zero, so that's, okay? The derivative of g would be 2x plus one, so the derivative of h would be f which is this the whole thing now times g prime. Plus f prime times g. Well that doesn't look quite the same but let's multiply it out and combine it. So putting these two together, I've got to x cubed, let's see. Multiply the first one plus x squared minus two up two times two is four, 4x minus 2. And then this one is plus 2x cubed plus 2x squared, let me combine my like terms. So I have both of these are the x cubed, so I have 4x cubed x squared terms. I've got one of those and two of those is three of those minus 4x minus 2, and lo and behold it is the same thing. So just by an example, I haven't proved to you that this is how it works. But this is the formula and sometimes you can see it written it doesn't matter, it's either you could write it as f times g prime plus g times f prime, if you would rather write it that way. Okay, so that is the derivative of a product. We come back and we'll talk about what the chain rule means.