Q. 5 - Calculus

Question

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

Step-by-Step Solution

Answer

$D_{u}f(x_0,y_0)=\underset{h\to 0}{\lim }\frac{f(x_0+a.h,y_0+b.h)-f(x_0,y_0)}{h}$

Step 1 : Introduction

The given is two variables $f(x,y)$.The objective is to find the definition of the directional derivative for a function of two variables,$f(x,y)$. The required definitions is provided

Step 2 :

Directional Derivative of a function of two variables:

Let $f(x,y)$ be a function of two variables defined on an open set containing the point $(x_0,y_0)$ and let $u=(a,b)$be a unit vector. The directional derivative of $f$ at $(x_0,y_0)$ in the direction of $u$,denoted by $D_{u}f(x_0,y_0)$ is given by,

$D_{u}f(x_0,y_0)=\underset{h\to 0}{\lim }\frac{f(x_0+a.h,y_0+b.h)-f(x_0,y_0)}{h}$

Provided that limit is exists.