 With two or more variables, we can also form what's known as mixed partial derivatives. And this will occur whenever we differentiate with one variable, and then differentiate with respect to a different variable. So our notation for that, if we differentiate with respect to x, then with respect to y, we'll designate this using our differential notation as... And we can also indicate this using subscript notation. And importantly, watch the order. With differential notation, the variable of differentiation goes from right to left, while with our function notation, the variable we're differentiating with respect to goes from left to right. As with differential notation, it helps to think about the derivative as an operator. It tells you to do something with whatever follows. So if we differentiate first with respect to x, then with respect to y, we'd write our function, differentiate with respect to x first, and then differentiate with respect to y. And that's why we write our differential notation this way. So let's find some mixed partial derivatives. So remember, this notation says we're going to differentiate with respect to x first and then with respect to y. So differentiating our function with respect to x will treat y as a constant. And really, this is the derivative of cosine. So the chain rule applies. That's minus sine times the derivative of the argument. Put things back where you found them. We need to find the derivative of this product. We'll apply the product rule. And since this is the partial derivative with respect to x, we're treating y as a constant. So the derivative of y with respect to x, well, y is a constant. So that's going to give us zero. And the derivative of x with respect to x, well, that's just one. And we can do a little cleanup. And now we need to differentiate minus y sine xy with respect to y. So this is a product, and we can apply the product rule. We have the partial derivative of sine xy with respect to y. So we're treating x like a constant. So our derivative will be... The derivative of minus y with respect to y will be... We have the derivative of this product xy, which will be... And again, since we're differentiating with respect to y, we're treating x like a constant. So this derivative will be zero. And so we get this as our final answer. What if we differentiate with respect to y first? So differentiating cosine xy with respect to y, and again, we're treating x like a constant. So this is the derivative of cosine. We have to differentiate the product. Since we're differentiating with respect to y, we treat x like a constant. And now we want to differentiate with respect to x. And after all the dust settles, we get this expression. And notice the mixed partial where we differentiated first with respect to x, and then with respect to y is the same as the mixed partial where we differentiated first with respect to y, and then with respect to x. Does this mean anything? Well, in our one example, we'll use our function notation for variety's sake. We found that fxy is equal to fyx. Therefore, from one example, we can conclude mixed partial derivatives are equal regardless of the order you take them in. I'd check this out and get a little bit more evidence than one example. But it does turn out to be true, provided we include some fine print. And the fine print is the following. Suppose that in some open region the mixed partials both exist and are continuous. Then, as long as these things are true, the mixed partials will be equal in the region.