Math Problem Statement

(1 point) The derivative of 𝑓(𝑥,𝑦,𝑧) at (0,3,9) is greatest in the direction of ⟨5,6,2⟩ . In that direction, the value of the derivative is 15 .

What is the derivative of 𝑓(𝑥,𝑦,𝑧) at (0,3,9) in the direction of ⟨−3,4,−1⟩ ?

Solution

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Math Problem Analysis

Mathematical Concepts

Multivariable Calculus
Directional Derivatives
Vector Calculus

Formulas

Directional Derivative: D_uf = ∇f · u
Unit Vector: u = v / |v|
Gradient Vector: ∇f

Theorems

Directional Derivative Theorem

Suitable Grade Level

Undergraduate Level (Calculus III)

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