 Hello, and welcome to this screencast on section 10.5, the chain rule. You are probably familiar with the term chain rule as one of the derivative rules you learned in single variable calculus. For the function f of two variables, x and y, where x and y are differentiable functions of an independent variable t, we have the following statement of the chain rule, which says that the derivative of z with respect to t is equal to the partial derivative of z with respect to x times the derivative of x with respect to t plus the partial derivative of z with respect to y times the derivative of y with respect to t. Note that the chain rule here is given in Leibniz notation, which makes it easier to keep track of the variables and the dependencies. And the statement of the chain rule here is very specific to the situation given, where z depends on x and y and x and y depend on t. We will not always have such a situation, nor will we always be considering situations with variables z, x, y, and t. For instance, we could have x and y depend on more than one quantity, or z is a function of more than two variables. To keep track of the chain rule in any given situation, we're going to introduce the notion of tree diagrams. Consider the familiar situation from the previous slide, where z depends on x and y, and x and y depend on t. To construct the tree diagram, we place the dependent variable z at the top of the tree and connect it to the variables for which it depends on one level below. So since z depends on x and y, we connect it to x and y one level below z. Since x and y both depend on t, then we connect both x and y to the variable t one level below. To now represent the chain rule, we label every edge of the diagram with an appropriate derivative or partial derivative. For instance, the edge from z to x is labeled with the partial derivative of z with respect to x. The edge from y to t is labeled with the derivative of y with respect to t. To calculate an overall derivative according to the chain rule, we construct the product of the derivatives along all paths connecting the variables and then add these products together. So for example, starting at the dependent variable z, we're going to trace down one path of the tree all the way to the bottom. Right, so we're going to trace this path. As we trace this path, we pass the partial derivative of z with respect to x and the derivative of x with respect to t. This gives us this term of the chain rule as the product of all the derivatives that we passed along that trace. We're going to do the same for all the other paths and for this tree, there's only one more path and that gives us this partial derivative times this derivative, which gives us this term. And we sum up everything, all the different traces that we make down the tree.