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The gradient vector is a powerful tool used in multivariable calculus. It's essentially a vector that stores all the partial derivative information of a function.

For a function of two variables, like our example with function \( f(x, y) = x^2 - xy + y^2 - y \), the gradient vector \( abla f(x, y) \) is expressed as \( (\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}) \). Each component of this vector represents the rate of change of the function in the respective dimension.

When you evaluate this gradient at a specific point, such as (1, -1) in our problem, you find the vector \( abla f(1, -1) \). This specific vector indicates the direction of steepest ascent of the function at that particular point.

Partial derivatives are used to understand how a function changes as one of its input variables varies, while the others are held constant.

In our function \( f(x, y) \), the partial derivative with respect to \( x \) is found by treating \( y \) as a constant and differentiating the equation just like you would a single-variable function. This gives us \( f_x = 2x - y \).

Similarly, to find the partial derivative with respect to \( y \), we treat \( x \) as a constant. This yields \( f_y = -x + 2y - 1 \).

The partial derivatives help us construct the gradient vector, indicating how the function behaves in various directions.

A unit vector is a vector that has a magnitude of one and points in the same direction as the original vector, essentially "normalizing" it.

To find a unit vector, you divide the vector by its magnitude. In our case, with the gradient vector at (1, -1) given by \( (3, -4) \), its magnitude is calculated as \( \sqrt{3^2 + (-4)^2} = 5 \).

The unit vector \( \mathbf{u} \) is calculated as \( \left( \frac{3}{5}, -\frac{4}{5} \right) \). This unit vector retains the direction of the gradient vector but scales it down to a length of one.

The directional derivative represents the rate at which the function changes as one moves in a specified direction, quantified by a unit vector.

It is computed as the dot product of the gradient vector and the unit vector: \( D_{\mathbf{u}} f(x, y) = abla f(x, y) \cdot \mathbf{u} \).

At the point (1, -1), the maximum directional derivative is equal to the magnitude of the gradient vector, which is 5. This happens when \( \mathbf{u} \) points in the same direction as \( abla f(1, -1) \). Conversely, the smallest (most negative) directional derivative, -5, occurs in the opposite direction.

To achieve specific directional derivative values like 4 or -3, the vector \( \mathbf{u} \) must be scaled to maintain orthogonality, ensuring the desired directional sensitivity when projected onto the gradient vector.