 We define the partial derivative as follows. Let f of x, y be a function of two variables. The partial derivative of f with respect to x, written this way, is going to be defined as a limit of... Well, it's a difference quotient, just like the definition of a regular derivative. And similarly, the partial derivative of f with respect to y is going to be the difference quotient, except this time y is going to be the variable that changes. Now, we also use the notation f subscript x to be partial of f with respect to x, and f subscript y. And we could similarly define the partial of f with respect to x, four functions of three variables, partial of f with respect to z, and so on. Now, since we restrict our limit process to one variable at a time, we say that we are holding constant the other variables. And to that end, it's helpful to remember what happens to limits and derivatives involving constants. So if c is a constant, one of our basic limit theorems is the limit of a constant is the constant, and the limit of constant times function is constant times the limit of the function. Meanwhile, if we differentiate a constant, we get zero, and the derivative of a constant times a function is going to be the constant times the derivative. And while we're at it, let's introduce that third limit process that we're familiar with. The integral of a constant is constant times variable plus some other constant, and the integral of constant times function is constant times the integral of the function. So for example, let f of x, y be x cubed minus e to the power x, y. Let's find the partial derivative of f with respect to x, the partial derivative of f with respect to y. So the crucial idea here is to find these partial derivatives between all other variables as constants. So to find the partial of f with respect to x, we treat all variables except x as constants. Now, it also helps to think about this in terms of our differential notation. This is the derivative with respect to x of x cubed minus e to the power x, y. So all of our regular derivative rules still apply because we're now thinking about y as a constant. So this is a function minus another function, and the derivative of a difference is the difference of the derivatives. This is the plain old derivative of x cubed, 3x squared. This is the derivative of e to the, and that's the world's simplest derivative. The derivative is going to be e to the times, don't forget the chain rule, the derivative of the exponent x, y. But x, y is a product, so this is the derivative of a product, and so we'll differentiate that. That's first times derivative second plus second times derivative first. So remember that we're treating all variables except for x as a constant. So this is the derivative of a constant, which is going to be 0. This is the derivative of x with respect to x, and that's going to be 1. And we'll do some algebraic simplification. And similarly, if we want to find the derivative of f with respect to y, we treat all variables except y as a constant. That's the derivative minus the derivative. Again, since we're differentiating with respect to y, all other variables are constants. So this is the derivative of a constant, which is going to be, this one's a little bit more complicated, this is the derivative of e to the, well that's the same thing times the derivative of the exponent. x, y is a product, so the derivative of a product is, and we know the derivative of y with respect to y. And again, since we're treating y as our only variable and all other variables as constants, this derivative is going to be the derivative of a constant. And we can clean up and get our final answer. We can find higher order derivatives in the same way we did in single variable calculus. So the second derivative of f with respect to x is designated this way, or fxx. The third derivative is designated this way, or fxxx. But the fourth derivative of f with respect to x is, well it's written the same way you would pretty much expect it to be written. And the only thing we have to keep track of here is remembering which is the variable and which is constant. So again, if it's not the variable we're differentiating with respect to, we'll treat it as a constant. So here we want to find the second derivative with respect to y of our function sine of x, y. Now it will be useful to recall that this comes from our operator notation. This second derivative really is saying, take the derivative with respect to y of the derivative with respect to y of our function sine of x, y. And again, parentheses mean the same thing they always do. Take care of the things inside. So let's find the derivative with respect to y of sine x, y. The chain rule is your friend. That's the derivative of sine, which is cosine, times the derivative of the argument. Since x, y is a product, the derivative of a product is, y is our variable. So this is the derivative of y with respect to y. That's just going to be 1. And x is not our variable. We have to treat it as a constant. So this derivative is going to be 0. And a little cleanup gets our answer, which is the derivative with respect to y of sine x, y. And now we need the derivative of x times cosine x, y. So that's a product. But remember, y is our variable and everything else we are treating as a constant. So x is a constant. And so the derivative of constant times function is constant times the derivative of the function. And again, the chain rule is your friend. This is the derivative of cosine, which is minus sine, times the derivative of the inside. That's a product, so we apply our product rule. x times derivative of y plus y times derivative of x. Again, the derivative of y with respect to y is going to be 1. And the derivative of x with respect to y is going to be 0. And we can clean up the algebra to get our final answer.