 Consider the surface z equals f of x, y at some point. If you go parallel to the x-axis, z changes at a rate equal to the partial of f with respect to x. If you go parallel to the y-axis, z changes at a rate equal to the partial of f with respect to y. But what if you go in some other direction? Then z changes at a rate equal to the directional derivative. So how do we find that directional derivative? Remember that slope is the ratio between the rise and the run. In three dimensions, the run is going to be measured on the x-y plane, and so we can treat it as a two-dimensional vector. Since the slope is a ratio, it's convenient to treat the run as having length one. So we'll want our vector to have magnitude one. Now suppose we have a vector with magnitude one. If we run in the direction parallel to the x-axis, then z will increase by a times the partial of z with respect to x. And that's because a is the amount by which x is changing, and that's going to be scaled by the rate of change of z with respect to x. And similarly, if we go in the direction parallel to the y-axis, then z will increase by b times the partial with respect to y, because y is increased by amount b, and the partial gives us the rate of increase. And so that means if we go in the direction ab, then z will increase by the sum of these two increases. And this is the directional derivative, which we indicate using the notation dvf. So for example, let z equals x squared minus y squared, how rapidly is z increasing at the point where x equals 5, y equals 3 if you go in the direction one, negative one. So we'll need to find the unit vector in the same direction, and so we'll scale our vector. First we'll find our magnitude, and so our unit vector will be. Now our directional derivative will need the partial of z with respect to x and the partial of z with respect to y, so we'll find those partial derivatives and substitute, which gives us our directional derivative. Since the directional derivative gives us the rate of change of z in different directions, we can ask in what direction should we go to increase z the most? Unfortunately, answering a question like that requires calculus. Oh, wait, you've already had two calculus courses before this one, so we should be able to do this. So I suppose you're at the point where x equals 3, y equals 2 on the graph of z equals x squared plus y squared in what direction is z increasing the most rapidly? Now it's helpful to note that the unit vector in the direction A, B can be expressed as cosine theta, sine theta. And so we can find our directional derivative, differentiating, and notice that our directional derivative is a function of theta. So if I want to find the maximum value, we can use calculus. So to find its maximum value, we'll differentiate. Now, ordinarily, our next step would be finding the critical value. Now, rather than trying to solve this, we might note that our direction vector itself is cosine theta, sine theta, but any scalar multiple of this will also go in the same direction. Now, since the equation for finding the critical value has 4 cosine theta, let's multiply our direction vector by 4 and get, but equals means replaceable. Since 4 cosine theta is equal to 6 sine theta, we can replace. And since both components have this factor of sine theta, we'll remove that factor. And again, a scalar multiple does not change the actual direction of the vector. So this means that the vector that will give us the greatest value of the directional derivative is going to be going in the direction 6, 4. To recap the logic, if we are going in the direction of greatest increase, then 6 sine of theta must be equal to 4 cosine theta. And since the direction we're going in is cosine theta sine theta, then this equality, 6 sine theta equals 4 cosine theta, means that we're going in some scalar multiple of the direction 6, 4. We can generalize this process. So again, suppose our directional derivative is given by... To find the maximum value, we'll differentiate with respect to theta. Our critical value we'll solve. And again, while our direction is expressed as cosine theta sine theta, any scalar multiple will go in the same direction. So if we multiply by our partial with respect to y, equals means replaceable. So this partial of f with respect to y cosine theta is really the same as the partial of f with respect to x of sine theta. We can remove this common factor of sine theta, and that scalar multiple doesn't affect the direction. So the direction is going to actually be partial of f with respect to x, partial of f with respect to y. And we know that this is the maximum value because our derivative is equal to 0. Well, we know we're at a critical value, but we might be at a minimum. So how do we know we're at a maximum and not a minimum? Well, let's consider, if we're moving in some direction, the unit vector will be some scalar multiple of that vector, where k is greater than 0. And so if we move in that direction, the directional derivative will be... But that's the sum of squares, so it must be positive. But what if we move in the exact opposite direction? If we move in the direction negative whatever k partial of f with respect to x, negative whenever k partial of f with respect to y, the directional derivative will be... which must be negative. Now, we know this unit vector corresponds to an extreme value, and we know there are other direction angles that give a lesser value of the directional derivative. And so this vector must correspond to a maximum. So if I want to find the direction of greatest increase of z on the graph of z equals x cubed minus y cubed where x equals negative 1, y equals 4, I'll find those partial derivatives. And x equals negative 1, 4. These partial derivatives have value. And the direction of greatest increase will occur in the direction of those partial derivatives. 3, negative 4 to 8.