 Hello, and welcome to this screencast on section 10.6, Directional Derivatives and the Gradient. The partial derivatives of a function tell us the instantaneous rate at which the function changes only in specific directions. In this screencast, we will introduce the directional derivative, which allows us to measure the rate at which a function changes in any direction. Jumping right in, the derivative of f at a point x, y in the direction of the unit vector u is denoted by d subscript u f of x, y and is given by the following limit. We insist on u being a unit vector so that this directional derivative measures the change in f per unit change in the direction of u. Just like we developed shortcut rules for standard derivatives and partial derivatives, we can also find a way to evaluate directional derivatives without resorting to this limit definition. Let's look at how to compute the directional derivative in general. Suppose we are interested in the instantaneous rate of change of f at a point x not, y not in the direction of a unit vector u. The variables x and y are changing according to these parameterizations, since we are changing by u sub 1 in the x direction and u sub 2 in the y direction. If we were to construct a tree diagram, we would have that the directional derivative depends on x, which then depends on t, and the directional derivative depends on y, which then depends on t. So by the chain rule, we can calculate the directional derivative using the following expression, and since the derivative of x with respect to t is equal to u sub 1 and the derivative of y with respect to t is equal to u sub 2, we get the following expression here at the bottom for the directional derivative. What we deduced on the previous slide is stated here. The directional derivative at a point x, y in the direction of a unit vector u may be computed by taking the partial of f with respect to x times the x component of the vector u, plus the partial derivative with respect to y times the y component of the vector u. Note that the directional derivative can be computed as the dot product of the following two vectors. The first vector has x component equal to the partial derivative with respect to x, and the y component equal to the partial derivative with respect to y. The second vector is u. It turns out that this first vector is special, as we will see in our work in the section. This vector is called the gradient of the function f, and we denote it using an upside down triangle next to f of x, y. The upside down triangle is the symbol called del. The gradient gives us an alternate way to state the directional derivative formula above, as we now replace here at the very top, in terms of the dot product of the gradient and the vector u.