The Directional Derivative
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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.
video by Dr Chris Tisdell 

As is common in multivariable 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.
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.
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.
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
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
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
video by Dr Chris Tisdell 

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

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Find \(D_u f(x,y)\) if we have the function \(f(x,y) = e^{x1}+e^y + 2x\) 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 \(f(x,y) = e^{x1}+e^y + 2x\) 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 \(f(x,y) = e^{x1}+e^y + 2x\) 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
video by Thomas Wernau 

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

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For the surface \( f(x,y) = 3x^3y^2+2xy^2 \) 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 \( f(x,y) = 3x^3y^2+2xy^2 \) 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
video by Thomas Wernau 

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