# Derivative of directional vector

• yecko
In summary, the conversation discusses finding the unit vectors along which given functions increase and decrease most rapidly at a given point, and then finding the derivatives of the functions in these directions. The directional derivative is found to be 2*sqrt(3) and the reason for this is explained. The conversation ends with the OP thanking for the help and realizing their mistake in calculating the gradient.
yecko
Gold Member

## Homework Statement

Find the unit vectors along which the given functions below increase and decrease most rapidly at P0 . Then find the derivatives of the functions in these directions.

solution:

## The Attempt at a Solution

why are the derivatives’ values along these two directions are 2*sqrt(3)?
as (∇f)(P0)= <1,1,1>, absolute value of it = sqrt(3)
why is the solution multiplied 2 for it?
thanks

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yecko said:
as (∇f)(P0)= <1,1,1>,
It isn’t. Double check your computations.

yecko said:

## Homework Statement

Find the unit vectors along which the given functions below increase and decrease most rapidly at P0 . Then find the derivatives of the functions in these directions.

View attachment 213475

## Homework Equations

solution:
View attachment 213474

## The Attempt at a Solution

why are the derivatives’ values along these two directions are 2*sqrt(3)?
as (∇f)(P0)= <1,1,1>, absolute value of it = sqrt(3)
why is the solution multiplied 2 for it?
thanks

You are correct: the directional derivative is, indeed, ##2 \sqrt{3}##. The reason is that ##\nabla f(1,1,1) = \langle 2,2,2 \rangle## is a vector of length ##\sqrt{3 \times 4} = 2 \sqrt{3}##.

Ray Vickson said:
You are correct: the directional derivative is, indeed, ##2 \sqrt{3}##.
The OP thinks the directional derivative is ##\sqrt{3}## and ##\nabla f = \langle 1,1,1\rangle##, so he is not correct. This is why I suggested him to check his computation of the gradient in #2.

o yes... thanks

## What is the definition of "derivative of directional vector"?

The derivative of a directional vector is the rate of change of the vector with respect to a specific input variable. It measures how much the vector changes in magnitude and direction as the input variable changes.

## How do you calculate the derivative of a directional vector?

To calculate the derivative of a directional vector, you first need to parametrize the vector in terms of a single variable. Then, you can use the chain rule and the product rule to find the derivative of each component of the vector. Finally, you can combine the derivatives to find the overall derivative of the directional vector.

## What is the physical interpretation of the derivative of a directional vector?

The physical interpretation of the derivative of a directional vector is the instantaneous rate of change of the vector at a specific point. It tells us how fast and in what direction the vector is changing at that point.

## What are some real-world applications of the derivative of directional vector?

The derivative of directional vector has many applications in physics, engineering, and economics. For example, it is used to calculate the velocity and acceleration of moving objects, the change in flow of fluids, and the rate of change of stock prices.

## How is the concept of derivative of directional vector related to other mathematical concepts?

The derivative of directional vector is closely related to other mathematical concepts such as vectors, limits, and derivatives of multivariable functions. It is also a key concept in vector calculus, which studies the properties and behaviors of vector fields.

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