 So remember the derivative is the rate of change with respect to some chosen variable. This remains true even in multivariable calculus with a minor change. If z equals f of x a function of a single variable, then dz dx is the rate of change of z with respect to x. But if z is a function of two variables that includes x, then the derivative is a rate of change with respect to x when we hold the other variables constant. But what if we can't? Suppose z equals f of x y, where y itself is a function of x. Then we can't find the partial of z with respect to x, since if we allow x to vary, y can't be held constant. However, if y is a function of x, then z is really a function of one variable, and we can find dz dx. But how? Well, let's see if we can find it with an actual function. Suppose z equals x squared plus y squared, where y is a function of x. Let's explain why we can't find the partial of z with respect to x, and then find dz dx. To find the partial derivative, we have to keep y constant. Except if y is a function of x, then changing x will change y. So it won't be possible to keep y constant, and finding the partial derivative is impossible. If we assume y is a function of x, we can apply the differential operator to find, and that gives us dz dx. And at this point, it's worth noticing that the first step in finding the derivative is differentiating an expression with respect to the variable in the expression. So we had to find the derivative of x squared with respect to x, but we also found the derivative of y squared with respect to y. That gave us the 2y. The important thing here is to notice that these are the same as the partial derivatives of our function with respect to x and with respect to y. We then multiply these by the single variable derivative of the variable with respect to x. And this leads to what we might call the multivariable chain rule. Suppose z is a function of several variables where the variables are assumed to be functions of some variable t, possibly among those already listed, or possibly something different. Then the derivative of z with respect to t is going to be the partial with respect to each of the variables times the derivative with respect to t. And it's important not to get too caught up in the formula. The multivariable chain rule is really just a formal statement of what you'd do anyway. For example, let z equals x cubed minus 3xy, we'll find dz dx. So if you'd never heard of the multivariable chain rule, we'd proceed as follows. We'll apply our differential operator with respect to x. And since dx dx is equal to 1, we can simplify. On the other hand, using this multivariable chain rule, that's going to be the partial with respect to x times the derivative of x with respect to x plus the partial with respect to y times the derivative with respect to x. And we'll find those partial derivatives, substitute. And again, since dx dx is equal to 1, this simplifies. And we get the same answer as before. Now it's worth applying a short consistency check because we have this newfangled method and we'd like to make sure that it gives us the same answers as he tried in true methods. So let's think about this. If z equals x cubed minus 3xy, we find dz dx. Now suppose z is equal to c, some constant, then this actually gives us the level curve in the plane z equal to c. Then implicit differentiation gives us, but if we're on the plane z equal to c, then dz dx is equal to 0. That's because dz dx is the rate of change of z, and if z is always equal to a constant, it's not changing, and so that rate of change is equal to 0. And what that means is the result we get from the multivariable chain rule and the result we get from implicit differentiation match up and they give us the same results. The other thing to check is to see what happens with partial derivatives. So again, if z equals x cubed minus 3xy, we found dz dx is equal to sub-expression. And our partial of z with respect to x will be... Now remember, we find the partial derivatives by assuming the other variables are held constant. And in this case, y is the other variable, so we found this partial derivative by assuming y constant. But if y is constant, dy dx, that's the rate of change of y with respect to x, is equal to 0. And again, the chain rule gives us the same result. So it's important to distinguish between the notation. So let f of x be this function of three variables. Let's find the partial of f with respect to x and df dx. So to find the partial of f with respect to x, we assume the other variables are constant, which gives us... To find df dx, we assume that all variables are functions of x. And so we'll take the partial with respect to each of x, y, and z, and then multiply by the derivative of each with respect to x. We'll fill in those partial derivatives. And again, since dx dx is equal to 1, we can simplify this to get our derivative with respect to x.