This page covers applications of gradients and directional derivatives.
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There are several applications where the gradient is used and has additional meanings based on what the function represents. Here are three of them.
Temperature
When the function f is a temperature, the gradient can be interpreted as the change in temperature.
Practice
If we have a material where the heat flow (from hot to cold) is given by \( T = x^3  3xy^2 \), determine the direction of maximum decrease of temperature at the point \( (1,2) \).
Problem Statement
If we have a material where the heat flow (from hot to cold) is given by \( T = x^3  3xy^2 \), determine the direction of maximum decrease of temperature at the point \( (1,2) \).
Solution
video by Dr Chris Tisdell 

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Tangent Plane
To determine the equation of a plane, we need one point and a normal vector. Since the gradient is normal to the level curves (and also to the tangent plane), we have a normal vector and we are usually given a point in the problem statement. This is enough information to determine the equation of the tangent plane.
Here is a video clip with a great explanation of this idea. He also explains how using the gradient notation simplifies the notation significantly.
video by Dr Chris Tisdell 

Practice
Calculate the tangent plane and the normal line to the surface \( x^2 + y^2 + z = 9 \) at the point \( (1,2,4) \).
Problem Statement
Calculate the tangent plane and the normal line to the surface \( x^2 + y^2 + z = 9 \) at the point \( (1,2,4) \).
Solution
video by Dr Chris Tisdell 

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Calculate the gradient, directional derivative and equation of the tangent plane of \( g(x,y,z) = \sin(xyz) \) at the point \( (\pi, 1/2, 1/2) \) in the direction \( \vec{v} = \langle \sqrt{3},2,3 \rangle \).
Problem Statement
Calculate the gradient, directional derivative and equation of the tangent plane of \( g(x,y,z) = \sin(xyz) \) at the point \( (\pi, 1/2, 1/2) \) in the direction \( \vec{v} = \langle \sqrt{3},2,3 \rangle \).
Solution
video by Dr Chris Tisdell 

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Calculate the equation of the tangent plane to \( x^4 + xy + y^2 = 19 \) at the point \( (2,3) \).
Problem Statement
Calculate the equation of the tangent plane to \( x^4 + xy + y^2 = 19 \) at the point \( (2,3) \).
Solution
video by Krista King Math 

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Calculate the equation of the tangent plane to the surface \(f(x,y)=2y\cos(5x3y)\) at the point \((3,5,10)\).
Problem Statement
Calculate the equation of the tangent plane to the surface \(f(x,y)=2y\cos(5x3y)\) at the point \((3,5,10)\).
Solution
video by MIP4U 

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Really UNDERSTAND Calculus
external links you may find helpful 

Better Explained: Vector Calculus: Understanding the Gradient 
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