# Directional derivative question

• julz127
In summary, the rate of change of f(x,y) = x/(1+y) in the direction (i-j) at the point (0,0) is given by Du = (y+x+1)/(y+1)^2, and the unit vector i-j should be used in the dot product with grad(f(x,y)).
julz127

## Homework Statement

rate of change of $f(x,y) = \frac{x}{(1+y)}$ in the direction (i-j) at the point (0,0)

## The Attempt at a Solution

$∇f(x,y) = \frac{1}{(y+1)}\hat{i} - \frac{x}{(y+1)^2}\hat{j}$

$D_u = ( f_x, f_y) \bullet ( 1, -1 )$

$D_u = \frac{(y+x+1)}{(y+1)^2}$

Wolfram and the answer sheet is telling me that there should be a $\sqrt{2}$ in the denominator, but I can't figure out where it comes from, thanks.

Last edited:
julz127 said:

## Homework Statement

rate of change of f(x,y) = x/(1+y) in the direction (i-j) at the point (0,0)

## The Attempt at a Solution

Du = ( f_x, f_y ) dot ( 1, -1 )

Du = (y+x+1)/(y+1)^2

Wolfram and the answer sheet is telling me that there should be a sqrt(2) in the denominator, but I can't figure out where it comes from, thanks.

The vector you want to dot the grad with should be a unit vector pointed in the direction i-j. That's what 'in the direction' means.

1 person

## What is a directional derivative?

A directional derivative is a measure of how a function changes in a specific direction from a given point. It is used to determine the rate of change of a function in a particular direction.

## How is a directional derivative calculated?

The directional derivative of a function is calculated using the dot product of the gradient vector of the function and a unit vector pointing in the desired direction. It can also be calculated using partial derivatives.

## Why is the directional derivative important?

The directional derivative is important because it allows us to determine the direction in which a function changes the fastest at a given point. This information is useful in various applications, such as optimization and physics.

## What is the difference between a directional derivative and a partial derivative?

A directional derivative is a measure of the rate of change of a function in a specific direction from a given point, while a partial derivative is a measure of the rate of change of a function with respect to one of its variables. A directional derivative can be calculated using partial derivatives, but the opposite is not always true.

## How is a directional derivative used in real life?

Directional derivatives are used in many fields, such as physics, engineering, and economics. For example, in physics, it can be used to determine the direction in which a particle will move in a magnetic field. In economics, it can be used to determine the direction of steepest ascent in a production function.

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