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The concept of a Gradient Vector is a crucial tool in multivariable calculus for understanding how a function changes at any point. A gradient vector is composed of partial derivatives of a multivariable function. In simpler terms, it represents a vector that points in the direction of the greatest rate of increase of the function.
For any given function, the gradient vector is defined as \( abla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right) \) for a function of three variables \( x, y, \text{and} z \). The gradient can also be thought of as a collection of all the first-order partial derivatives of the function.
An important characteristic of the gradient vector is its ability to determine the rate of change of the function in any given direction. The computation of this vector involves taking derivatives with respect to every variable present in the function, revealing how the function behaves in different directions.

Partial derivatives are used when you have a function of multiple variables and you want to understand how the function changes when you alter one variable while keeping the others constant. This concept is foundational to finding the gradient vector.
For a function \( f(x, y, z) = yz2^x \), the partial derivative with respect to \( x \) would treat \( y \) and \( z \) as constants, deriving the effect of \( x \) alone on the function. Similarly, you compute partial derivatives with respect to \( y \) and \( z \).

• \( \frac{\partial f}{\partial x} = yz \ln(2) \cdot 2^x \)
• \( \frac{\partial f}{\partial y} = z2^x \)
• \( \frac{\partial f}{\partial z} = y2^x \)

These expressions provide a view into how the function changes along each axis, contributing to the overall gradient vector.
Evaluating these derivatives at a given point gives specific information about the rate and direction of change at that point.

The process to Normalize a Vector involves scaling a vector so that its length or magnitude becomes 1, while maintaining its direction. This is essential in cases where the direction is important, but the magnitude needs to be standardized, as is the case with directional derivatives.
For a vector \( \mathbf{a} = 2 \mathbf{j} - \mathbf{k} \), the normalization process first calculates the magnitude \( \| \mathbf{a} \| \), which is obtained using the formula \( \sqrt{2^2 + (-1)^2} = \sqrt{5} \).
Then, the normalized vector is:

• \( \frac{\mathbf{a}}{\| \mathbf{a} \|} = \left( 0, \frac{2}{\sqrt{5}}, -\frac{1}{\sqrt{5}} \right) \)

This normalized vector preserves direction but adjusts its magnitude to simplify the computation of directional derivatives by converting the vector into a unit vector.

The Dot Product is a method to multiply two vectors resulting in a scalar. It measures the extent to which two vectors align with each other, and is particularly useful in finding the directional derivative.
The formula for a dot product of two vectors \( \mathbf{A} = (a_1, a_2, a_3) \) and \( \mathbf{B} = (b_1, b_2, b_3) \) is given by \( a_1b_1 + a_2b_2 + a_3b_3 \). This dot product highlights how much of one vector goes in the direction of another.
In the exercise, the dot product of the gradient at point \( P \) \( (-2\ln(2), 2, -2) \) and the normalized direction vector \( \left( 0, \frac{2}{\sqrt{5}}, -\frac{1}{\sqrt{5}} \right) \) is computed:

• \( -2\ln(2) \times 0 + 2 \times \frac{2}{\sqrt{5}} + (-2) \times \left(-\frac{1}{\sqrt{5}} \right) = \frac{6}{\sqrt{5}} \)

This result shows the rate of change of the function at point \( P \) in the direction of the given vector \( \mathbf{a} \).
The dot product thus forms an essential step in calculating the directional derivative.