What are Directional Derivatives

What are directional derivatives ● Directional derivative is a concept in probability theory, and more specifically in stochastic calculus. It is the time derivative of the expected value with respect to thedirection of motion of the Brownian motion process. It can be used to define the rateof change of mean reversion in option pricing models. ● The directional derivative is the first derivative of a function that is not only differentiable, but also has a non-zero value in some direction. Directional derivatives, here we go. Let us assume that we have a function of two variables, x and y. We can compute partial derivatives of a function with respect to x or y, which measure howthe value of the function changes if we move in a particular direction (x-axis or y-axis). Butwhat about moving in other directions? Of course, we have seen approximate formulas and so on. But we can still ask if there is aderivative in every direction, and that's actually yes. That is what is called the directionalderivative. The derivatives in the direction of i-hat or j-hat are vectors that go along the x or y-axis. So what if we move in another direction? Let's say the direction of some unit vector, u. If I give you a unit vector, you can ask yourself how quickly the function changes if youmove in that direction. Let's look at the straight-line trajectory. In this model, I start at some value (x, y). Next, Ihave a vector (u) and I'm going to move in a straight line in the direction of u. Finally, Ihave the graph of my function. And we ask ourselves, how quickly does the value change when I move on the graph inthat direction.