Math Problem Statement
Find the gradient of the function F(x,y,z)=xy3z⎯⎯⎯⎯⎯√ at the point (−6,7,7) . Write the exact answer. Do not round.
Solution
To find the gradient of the function at the point , we need to compute the partial derivatives with respect to , , and and then evaluate them at the given point.
Step 1: Compute the partial derivatives
The function is . Let's find each of the partial derivatives:
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Partial derivative with respect to :
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Partial derivative with respect to :
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Partial derivative with respect to :
Step 2: Evaluate at the point
Now, we substitute the values , , and into the partial derivatives:
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Evaluate at :
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Evaluate at : [ \frac{\partial F}{\partial y} = x \cdot 3y^2 \sqrt{z} = (-6) \cdot 3 \cdot 7^2 \cdot \sqrt{7} =
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Math Problem Analysis
Mathematical Concepts
Multivariable Calculus
Gradient
Partial Derivatives
Formulas
Gradient of a function: ∇F = (∂F/∂x, ∂F/∂y, ∂F/∂z)
Partial derivatives of F(x, y, z) = x y^3 √z
Theorems
Chain rule for partial derivatives
Suitable Grade Level
Grades 11-12
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