The Gradient

Definition (Gradient) Let be a differentiable function at and let have partial derivatives and Then the gradient of is denoted by



The value of the gradient at the point is denoted by

Proposition (Gradient) Let and be differentiable functions. Then

        (i) for any constant
        
        (ii) for any constants and
        
        (iii)
        
        (iv)   provided
        
        (v)
        
        
    Proof. The Constant Rule is proved as follows:

  

The Linearity Rule is proved as follows:







The Product Rule is proved as follows:











The Quotient Rule is proved as follows:











The Power Rule is proved as follows:

Example (Gradient) Find the gradient of the function,

    Solution. If then



and

Recommended Reading
functions of several variables
graphs of functions
polynomial functions
rational functions
level curves
level surfaces
limits of multivariate functions
continuity of multivariate functions
partial derivatives
higher order partial derivatives
tangent planes
total differential
linear approximation with multivariate functions
differentiability
chain rule with one independent parameter
chain rule with two independent parameters
chain rule with several independent parameters
directional derivatives
the gradient
the gradient and directional derivatives
steepest ascent and steepest descent
normal property of the gradient
tangent planes and normal lines
relative extrema
critical points
second partials test
absolute extrema
lagrange multipliers with one parameter
lagrange multipliers with two parameters

Recommended Math Books
Thomas' Calculus, Early Transcendentals, Media Upgrade (11th Edition)
Thomas' Calculus, Media Upgrade (11th Edition)
Thomas' Calculus Early Transcendentals; Student's Solutions Manual; Part One
Calculus (With Analytic Geometry)(8th edition)
Calculus (Stewart's Calculus Series)
Applied Calculus
Calculus Textbooks
Elementary Calculus
Advanced Calculus
Supplementary Resources

Recommended Math Gifts
Math Happy
Calculus Happy
Homework Happy
Limits Happy
I Love Math
I Love Calculus
I Love Homework
I Love Multivariate Calculus
I Love Limits

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