The directional derivative of f(x, y, z) = xyz at point (-1, 1, 3) in the direction of vector i - 2j + 2k isa)3i - 3j - kb)−7/3c)7/3d)7Correct answer is option 'C'. Can you explain this answer?

Question

The directional derivative of f(x, y, z) = xyz at point (-1, 1, 3) in the direction of vector i - 2j + 2k isa)3i - 3j - kb)&minus;7/3c)7/3d)7Correct answer is option 'C'. Can you explain this answer?

Answer

Concept:

Directional Derivative = Gradient of function × Unit direction Vector

If F = f(x, y, z) then,

For the given direction vector

Unit direction vector =

and

Calculation:

Given, f(x,y,z) = xyz

∴ (Grad f)_P =

Direction vector

Unit direction vector

=

∴ Directional Derivative =

D. D =

Answer

Calculating the Directional Derivative:

To calculate the directional derivative of the function $ f(x, y, z) = xyz $ at the point (-1, 1, 3) in the direction of the vector $ \textbf{i} - 2\textbf{j} + 2\textbf{k} $, we follow these steps:

1. Find the Gradient of the Function:
The gradient of a function is given by the vector $ \nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right) $.
For the function $ f(x, y, z) = xyz $, the gradient is $ \nabla f = \left( yz, xz, xy \right) $.

2. Compute the Gradient at the Given Point:
Substitute the values of x, y, and z from the given point (-1, 1, 3) into the gradient vector to get $ \nabla f = \left( 1 \times 3, -1 \times 3, -1 \times 1 \right) = (3, -3, -1) $.

3. Normalize the Direction Vector:
Normalize the given direction vector $ \textbf{v} = \textbf{i} - 2\textbf{j} + 2\textbf{k} $ to get a unit vector $ \textbf{u} $ in the same direction.

4. Calculate the Directional Derivative:
The directional derivative is given by the dot product of the gradient and the unit direction vector:
$ D_{\textbf{u}} f = \nabla f \cdot \textbf{u} = (3, -3, -1) \cdot \left( \frac{1}{3}, -\frac{2}{3}, \frac{2}{3} \right) = 7/3 $.

Therefore, the directional derivative of $ f(x, y, z) = xyz $ at point (-1, 1, 3) in the direction of the vector $ \textbf{i} - 2\textbf{j} + 2\textbf{k} $ is $ 7/3 $ (option C).

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