 Hello, and welcome to this screencast on section 10.2, First Order Partial Derivatives. The derivative plays a central role in first semester calculus because it provides important information about a function. Thinking graphically, for instance, the derivative at a point tells us the slope of the tangent line to the graph at that point. In addition, the derivative at a point also provides the instantaneous rate of change of the function with respect to changes in the independent variable. Now that we are investigating functions of two or more variables, how can we use this notion of derivative? It turns out that we can still ask how fast the function is changing, though we have to be careful about what we mean. Thinking graphically, we can try to measure how steep the graph of a function is in a particular direction. Recall in chapter 9, we studied the behavior of a function of two or more variables by considering the traces of a function. A trace helps us understand how the function is changing in a particular direction by holding either x or y constant. Pictured here in red is a trace that holds x constant, and pictured here is a trace that holds y constant. Just as traces help us understand how a function changes in a particular direction, we can use derivatives to help us understand how fast the function is changing in these directions. The first order partial derivative of f with respect to x at a point a, b is defined by the following limit. This is the first derivative of f with respect to x while holding y constant. Similarly, we define the first order partial derivative of f with respect to y at a point a, b using this limit. This is the first derivative of f with respect to y while holding x constant. Each partial derivative at a point arises as the derivative of a one variable function defined by fixing one of the coordinates. Thus, computing partial derivatives is straightforward. We use the standard rules of single variable calculus, but do so while holding one or more of the variables constant. For example, consider this function f. To obtain the partial derivative of f with respect to x, we treat y as a constant and take the derivative of the expression with respect to x alone. In doing this, we get this as the first order partial derivative of f with respect to x. Similarly, to obtain the first partial derivative of f with respect to y, we treat x as constant and take the derivative of the expression with respect to y. This gives us this partial derivative. Note that the derivative of the term 5x in the function f is equal to 0 since we are treating x as constant. Due to the connection between one variable derivatives and partial derivatives, we will often use Leibniz style notation to denote partial derivatives as denoted here. Again, this is just another notation for the first order partial derivatives that we defined in this screencast. To see the contrast between the familiar Leibniz notation for single variable derivatives and now for partial derivatives, we can see this here, comparing the two notations side by side. The difference in notation helps us understand what sort of derivative we are looking for.