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17Calculus Partial Derivatives - The Directional Derivative

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The Directional Derivative

Topics You Need To Understand For This Page

basics of vectors partial derivatives gradient vector

From the gradient vector page you know that the gradient is a vector that contains information about the slope at every point. The directional derivative is the magnitude of the gradient in a certain direction. Here is a great a clip from a video explaining the directional derivative and gradient in detail and the associated notation, including some examples.

Dr Chris Tisdell - Gradient and Directional Derivative (Part 1) [19mins-21secs]

video by Dr Chris Tisdell

As is common in multi-variable calculus, we use specific notation to communicate concepts. Here is the notation we use to specify the directional derivative.

\(\displaystyle{ D_{\vec{u}}f = \nabla f \cdot \vec{u} }\)
We read this as the directional derivative of \(f\) in the direction of \(\vec{u}\).

The directional derivative can be thought of as the magnitude of the gradient at a certain point in a specific direction. To calculate the directional derivative, we take the dot product of the gradient with the unit vector ( \(\vec{u}\) ) in the direction we are interested in. The equation looks like this.

Here is a video going through a proof showing that the directional derivative is the dot product of the gradient with a direction vector. This video is not required for understanding how to use the dot product version of the directional derivative but it will help you understand it. So it is a good video to watch.

Dr Chris Tisdell - Proof: Directional Derivative is a Dot Product [8mins-6secs]

video by Dr Chris Tisdell

Here is a video clip discussing the properties of the directional derivative based on the use of the dot product.

Dr Chris Tisdell - Gradient and Directional Derivative (Part 2) [13mins-23secs]

video by Dr Chris Tisdell

From the last video clip, you now know that the direction of maximum increase of a function \(f\) is the gradient \(\nabla f\) and the direction of maximum decrease is \(-\nabla f\). (Remember that these are vectors.)

Since the gradient is just a partial derivative, the basic rules of algebra are followed as well as the chain rule. If you would like to confirm these rules, here is a quick video clip to watch.

Dr Chris Tisdell - Gradient and Directional Derivative (Part 3) [2mins-25secs]

video by Dr Chris Tisdell

Okay, time for some practice problems.

Practice

Unless otherwise instructed, find the directional derivative of these functions.

Find the change in \(f(x,y)=x^2y^3\) in the direction of \(\vec{v}=\hat{i}+\hat{j}\) at the point \((-2,1)\)

Problem Statement

Find the change in \(f(x,y)=x^2y^3\) in the direction of \(\vec{v}=\hat{i}+\hat{j}\) at the point \((-2,1)\)

Solution

Thomas Wernau - 4350 video solution

video by Thomas Wernau

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Calculate the derivative of \( f(x,y) = xe^y + \cos(xy) \) at the point \( (2,0) \) in the direction of \( \vec{v} = 3\hat{i} - 4\hat{j} \).

Problem Statement

Calculate the derivative of \( f(x,y) = xe^y + \cos(xy) \) at the point \( (2,0) \) in the direction of \( \vec{v} = 3\hat{i} - 4\hat{j} \).

Solution

Dr Chris Tisdell - 799 video solution

video by Dr Chris Tisdell

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At the point \( (1,1) \), determine the directions in which \( f(x,y) = x^2/2 + y^2/2 \) (a) increases most rapidly; (b) decreases most rapidly; (c) has zero change

Problem Statement

At the point \( (1,1) \), determine the directions in which \( f(x,y) = x^2/2 + y^2/2 \) (a) increases most rapidly; (b) decreases most rapidly; (c) has zero change

Solution

Dr Chris Tisdell - 800 video solution

video by Dr Chris Tisdell

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Calculate the derivative of \( f(x,y,z) = x^3 - xy^2-z \) at the point \( (1,1,0) \) in the direction \( \vec{v} = 2\hat{i} - 3\hat{j} + 6\hat{k}\).

Problem Statement

Calculate the derivative of \( f(x,y,z) = x^3 - xy^2-z \) at the point \( (1,1,0) \) in the direction \( \vec{v} = 2\hat{i} - 3\hat{j} + 6\hat{k}\).

Solution

Dr Chris Tisdell - 801 video solution

video by Dr Chris Tisdell

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Calculate the directional derivative of \( f(x,y) = 1 - x^2/2 - y^4/4 \) in the direction \(\hat{i} + \hat{j}\) at the point \( (1,2) \).

Problem Statement

Calculate the directional derivative of \( f(x,y) = 1 - x^2/2 - y^4/4 \) in the direction \(\hat{i} + \hat{j}\) at the point \( (1,2) \).

Solution

Dr Chris Tisdell - 803 video solution

video by Dr Chris Tisdell

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Calculate the directional derivative of \( f(x,y) = x^2 + 2xy + 3y^2 \) at the point \( (2,1) \) in the direction \(\hat{i}+\hat{j}\).

Problem Statement

Calculate the directional derivative of \( f(x,y) = x^2 + 2xy + 3y^2 \) at the point \( (2,1) \) in the direction \(\hat{i}+\hat{j}\).

Solution

Krista King Math - 807 video solution

video by Krista King Math

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Calculate the maximum directional derivative and its direction for \( f(x,y) = 2x^2 + 3xy + 4y^2 \) at \( (1,1) \).

Problem Statement

Calculate the maximum directional derivative and its direction for \( f(x,y) = 2x^2 + 3xy + 4y^2 \) at \( (1,1) \).

Solution

Krista King Math - 808 video solution

video by Krista King Math

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Calculate the directional derivative of \( f(x,y) = x^2y^3 - y^4 \) at \( (2,1) \) in the direction given by the angle \( \theta = \pi/4 \).

Problem Statement

Calculate the directional derivative of \( f(x,y) = x^2y^3 - y^4 \) at \( (2,1) \) in the direction given by the angle \( \theta = \pi/4 \).

Solution

PatrickJMT - 810 video solution

video by PatrickJMT

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Calculate the value of the maximum rate of change of the function \( f(x,y) = \sin(xy) \) at the point \( (1,0) \).

Problem Statement

Calculate the value of the maximum rate of change of the function \( f(x,y) = \sin(xy) \) at the point \( (1,0) \).

Solution

Krista King Math - 1522 video solution

video by Krista King Math

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Calculate the direction of the maximum rate of change of the scalar field \( \phi(x,y,z) = -2xy + x\ln(y+z) \) at \( P(1,3,-2) \) and the maximum value of the directional derivative at P.

Problem Statement

Calculate the direction of the maximum rate of change of the scalar field \( \phi(x,y,z) = -2xy + x\ln(y+z) \) at \( P(1,3,-2) \) and the maximum value of the directional derivative at P.

Solution

Dr Chris Tisdell - 1776 video solution

video by Dr Chris Tisdell

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Find the directional derivative \( D_x f(x,y) \) of the function \( f(x,y) = x^2 - 4y^2 + 2x \) at the point \((1,1)\) in the direction of the vector \( (1/2)\hat{i} - (\sqrt{3}/2)\hat{j} \)

Problem Statement

Find the directional derivative \( D_x f(x,y) \) of the function \( f(x,y) = x^2 - 4y^2 + 2x \) at the point \((1,1)\) in the direction of the vector \( (1/2)\hat{i} - (\sqrt{3}/2)\hat{j} \)

Solution

Thomas Wernau - 4335 video solution

video by Thomas Wernau

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Find the directional derivative \( D_u f(x,y) \) of the function \( f(x,y) = e^{xy}+y \) at \((2,0)\) in the direction of \(-\hat{i} + \hat{j}\sqrt{3}\)

Problem Statement

Find the directional derivative \( D_u f(x,y) \) of the function \( f(x,y) = e^{xy}+y \) at \((2,0)\) in the direction of \(-\hat{i} + \hat{j}\sqrt{3}\)

Solution

Thomas Wernau - 4336 video solution

video by Thomas Wernau

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Write the directional derivative \( D_u f(x,y) \) for the surface \( f(x,y) = xy^2 - 4x + 3y \) in the direction of \( 3\hat{i} - 4\hat{j} \)

Problem Statement

Write the directional derivative \( D_u f(x,y) \) for the surface \( f(x,y) = xy^2 - 4x + 3y \) in the direction of \( 3\hat{i} - 4\hat{j} \)

Solution

Thomas Wernau - 4337 video solution

video by Thomas Wernau

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Find \(D_u f(x,y)\) if we have the function \(f(x,y) = e^{x-1}+e^y + 2x\) and are at the point where \(x=1\) and \(y=0\) and want to move in the direction where \(\theta = 7\pi / 6\)

Problem Statement

Find \(D_u f(x,y)\) if we have the function \(f(x,y) = e^{x-1}+e^y + 2x\) and are at the point where \(x=1\) and \(y=0\) and want to move in the direction where \(\theta = 7\pi / 6\)

Hint

To get the unit vector in the direction of motion from the angle, use \( \vec{u} = \langle \cos \theta, \sin \theta \rangle \)

Problem Statement

Find \(D_u f(x,y)\) if we have the function \(f(x,y) = e^{x-1}+e^y + 2x\) and are at the point where \(x=1\) and \(y=0\) and want to move in the direction where \(\theta = 7\pi / 6\)

Hint

To get the unit vector in the direction of motion from the angle, use \( \vec{u} = \langle \cos \theta, \sin \theta \rangle \)

Solution

Thomas Wernau - 4352 video solution

video by Thomas Wernau

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Find \(D_u f(x,y)\) if we have the function \(f(x,y) = xy^3-x^2\) and are at the point where \(x=1\) and \(y=0\) and we want to move toward the origin.

Problem Statement

Find \(D_u f(x,y)\) if we have the function \(f(x,y) = xy^3-x^2\) and are at the point where \(x=1\) and \(y=0\) and we want to move toward the origin.

Solution

Thomas Wernau - 4353 video solution

video by Thomas Wernau

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a. Find \(\nabla f(x,y)\) if we have the function \(f(x,y) = x^2y^2 - x^2+y^4 - y\).
b. Find \(\nabla f(1,0)\) from the same function.
c. Write the directional derivative \(D_uf(x,y)\) as the dot product of the function at \((1,0)\) and in the direction of the vector \(\langle -5, 12 \rangle\).

Problem Statement

a. Find \(\nabla f(x,y)\) if we have the function \(f(x,y) = x^2y^2 - x^2+y^4 - y\).
b. Find \(\nabla f(1,0)\) from the same function.
c. Write the directional derivative \(D_uf(x,y)\) as the dot product of the function at \((1,0)\) and in the direction of the vector \(\langle -5, 12 \rangle\).

Solution

Thomas Wernau - 4354 video solution

video by Thomas Wernau

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Take the given function and direction vector and convert the directional derivative \( D_u f(x,y) \) into a dot product.
\( f(x,y) = e^{xy} + y \) in the direction of \(-\hat{i} + \hat{j}\sqrt{3}\)
Write the appropriate notation for the directional derivative.

Problem Statement

Take the given function and direction vector and convert the directional derivative \( D_u f(x,y) \) into a dot product.
\( f(x,y) = e^{xy} + y \) in the direction of \(-\hat{i} + \hat{j}\sqrt{3}\)
Write the appropriate notation for the directional derivative.

Solution

Thomas Wernau - 4338 video solution

video by Thomas Wernau

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Given a flies motion through space defined by the equation \( f(x,y) = xy^2 -4x + 3y \), solve these problems.
a) If it's location is at the point \((1,1,0)\) and it wants to move toward the point \((1,0,-4)\) find the steepness of its flight.
b) If it arrives at the point \((1,0,-4)\) and wants to move back to the origin, find the steepness of the flight at that instant.
c) If the fly arrives at the origin and wants to move upward at the greatest possible rate, what direction should it go and what will the steepness be?

Problem Statement

Given a flies motion through space defined by the equation \( f(x,y) = xy^2 -4x + 3y \), solve these problems.
a) If it's location is at the point \((1,1,0)\) and it wants to move toward the point \((1,0,-4)\) find the steepness of its flight.
b) If it arrives at the point \((1,0,-4)\) and wants to move back to the origin, find the steepness of the flight at that instant.
c) If the fly arrives at the origin and wants to move upward at the greatest possible rate, what direction should it go and what will the steepness be?

Solution

Thomas Wernau - 4339 video solution

video by Thomas Wernau

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For the surface \( f(x,y) = 3x^3-y^2+2xy^2 \) find the following
a. Find the complete point when \(x=1\) and \(y=0\).
b. Find the partial derivatives of \(f\)
c. Find the gradient function \( \nabla f(x,y)\)
d. Find the equation for the directional derivative in the direction of the vector \( \langle \cos \theta, \sin \theta \rangle \) at the point found in part a.
e. If a particle is at point a and moves toward the point \((1,3,1)\) what will \(D_u f(x,y)\) be?
f. In what direction should the particle move if it wants \( D_uf(x,y) \) to be the smallest value possible?

Problem Statement

For the surface \( f(x,y) = 3x^3-y^2+2xy^2 \) find the following
a. Find the complete point when \(x=1\) and \(y=0\).
b. Find the partial derivatives of \(f\)
c. Find the gradient function \( \nabla f(x,y)\)
d. Find the equation for the directional derivative in the direction of the vector \( \langle \cos \theta, \sin \theta \rangle \) at the point found in part a.
e. If a particle is at point a and moves toward the point \((1,3,1)\) what will \(D_u f(x,y)\) be?
f. In what direction should the particle move if it wants \( D_uf(x,y) \) to be the smallest value possible?

Solution

Thomas Wernau - 4351 video solution

video by Thomas Wernau

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Practice Instructions

Unless otherwise instructed, find the directional derivative of these functions.

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