So, plus g of x, plus g of x, times this f of x plus h, times f of x plus h, minus this f of x, minus that f of x, all of that over h, all of that over h. Now, we know from our limit properties, the limit of all of this business, well, that's just going to be the same thing as the limit of this as h approaches zero, plus the limit of this as h approaches zero, and then the limit of the product is going to be the same thing as the product of the limits. So, if I use both of those limit properties, I can rewrite this whole thing as the limit, let me give myself some real estate, the limit as h approaches zero of f of x plus h, of f of x plus h, times, times the limit as h approaches zero of all of this business, g of x plus h, minus g of x, minus g of x, all of that over h, I think you might see where this is going, very exciting, all right, plus, plus the limit, let me write that a little bit more clearly, plus the limit as h approaches zero of g of x, our nice brown colored g of x, times, now we're going to have a product here, the limit, the limit as h approaches zero of f of x plus h, of f of x plus h, minus f of x, minus f of x, all of that, all of that over h, and let me put the parentheses where they're appropriate, so that, that, that, that, and all I did here, the limit, the limit of this sum, that's going to be the sum of the limits, that's going to be the limit of this, plus the limit of that, and then the limit of the products is going to be the same thing as the product of the limits, so I just use those limit properties here, but now let's evaluate them. What's the limit, and I'll do it in different colors, what's this thing right over here? The limit as h approaches zero of f of x plus h, well that's just going to be f of x. Now, and this is the exciting part, what is this? The limit as h approaches zero of g of x plus h minus g of x over h. Well that's just our, that's the definition of our derivative, that's the derivative of g, so this is going to be, this is going to be the derivative of g of x, which is going to be g prime of x, g prime of x, so you're multiplying these two, and then you're going to have plus, what's the limit as h approaches zero of g of x? Well there's not even any h in here, so this is just going to be g of x, so plus g of x times the limit, so let's see, this one is in brown, and the last one I'll do in yellow, times the limit as h approaches zero, and we're getting very close, deserve, the drum roll should be starting, limit as h approaches zero of f of x plus h minus f of x over h. Well that's the definition of the derivative of f of x, this is f prime of x times f prime of x, so there you have it, the derivative of f of x times g of x is this, and if I wanted to write it in a little bit more condensed form, it is equal to, it is equal to f of x times the derivative of g with respect to x, times the derivative of g with respect to x plus g of x, plus g of x times the derivative of f with respect to x, f with respect to x, or another way to think about it, this is the first function times the derivative of the second, plus the second function times the derivative of the first, this is the proof, or a proof, there's actually others, of the product rule.