Q. 2 - Calculus

Question

What is the definition of the directional derivative for a function of three variables, $f(x, y, z)$ ? Be sure to include the words "unit vector" in your definition.

Step-by-Step Solution

Answer

Going to assume that limit exists is $D_{u}f\left(x_0,y_0,z_0\right)={Lim}_{h\to 0}\frac{f\left(x_0+a·h,y_0+b·h,z_0+c·h\right)-f\left(x_0,y_0,z_0\right)}{h}$

Step1: Unit vector.

A vector could be a quantity that has both a magnitude and a direction related to it.
A unit vector could be a vector with a magnitude of $1$
It's also observed as a Direction Vector. 

Step2: Directional Derivative

Directional Derivative of a function of three variables:

Let$f(x, y, z)$ be a function of two variables defined on an open set containing the point $\left(x_0,y_0,z_0\right)$, and let $u=(a, b, c)$ be a unit vector.

The directional derivative with in $f$at $\left(x_{o,}{y}_{o,}{z}_o\right)$ in the direction of $u$, denoted by

$D_{u}f(x_0,y_0,z_o)$, is given by;

$D_{u}f\left(x_0,y_0,z_0\right)={Lim}_{h\to 0}\frac{f\left(x_0+a·h,y_0+b·h,z_0+c·h\right)-f\left(x_0,y_0,z_0\right)}{h}$

Provided that a limit existed