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The concept of the gradient is essential to understanding directional derivatives in multivariable calculus. When we talk about **gradient**, we refer to a vector that indicates both the direction and rate of the steepest ascent of a scalar field. For example, consider a function of two variables, such as our exercise's function \( f(x, y) = x^2 + y^2 \).
The gradient of \( f \) is denoted as \( abla f \). To find this vector, you'll need to calculate the partial derivatives of the function concerning each variable involved. In this case, determine \( \frac{\partial f}{\partial x} = 2x \) and \( \frac{\partial f}{\partial y} = 2y \).

After evaluating these partial derivatives, you form the gradient vector:

• \( abla f = (2x, 2y) \)

By inserting the point \( \mathbf{a} = (1, 2) \) into the gradient, the resulting vector is \( (2, 4) \).
The gradient not only shows the direction of greatest increase but also provides the rate of that increase. In essence, gradients are crucial because they're like guides to finding the steepest path of elevation in a landscape.

Understanding **partial derivatives** is fundamental for grasping how functions of multiple variables change. Unlike regular derivatives, which consider changes along a single axis, partial derivatives focus on the change of a function as one variable varies, holding others constant.
In our scenario, we look at the function \( f(x, y) = x^{2} + y^{2} \). To compute the partial derivative concerning \( x \), treat \( y \) as a constant. Thus, calculate:

• \( \frac{\partial f}{\partial x} = 2x \)

Similarly, for \( y \), hold \( x \) constant and find:

• \( \frac{\partial f}{\partial y} = 2y \)

Through these partial derivatives, you understand how each individual variable affects the overall function.

This is particularly practical in many fields like economics, physics, and engineering, where systems often involve multiple changing factors. It allows us to analyze how small changes in one dimension contribute to changes in the function's value.

A **unit vector** is a vector that has a magnitude of 1, allowing it to represent direction without affecting size or other properties of the resulting vector operation.
In the context of directional derivatives, converting the given direction vector \( \mathbf{v} = (-2, 1) \) into a unit vector \( \mathbf{u} \) is essential to measure "pure" directionality.

Begin by determining the norm (or magnitude) of \( \mathbf{v} \):

• \(||\mathbf{v}|| = \sqrt{(-2)^2 + 1^2} = \sqrt{4 + 1} = \sqrt{5}\)

Then, to find the unit vector, divide each component of \( \mathbf{v} \) by this magnitude:

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

This transformation ensures we analyze only the direction while engaging with the directional derivative.
Using unit vectors helps in standardizing and simplifying various calculations across many applications, ensuring similar conditions irrespective of the vector's original length.