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Partial derivatives are a fundamental concept in calculus for functions of several variables. When you have a function like \( f(x, y) = x^2 - 3xy + 4y^2 \), the partial derivative allows us to see how the function changes as we vary one variable while keeping the others constant. This is akin to taking a slice of the function and examining how its shape changes in a particular direction.

• The partial derivative with respect to \( x \) is denoted as \( \frac{\partial f}{\partial x} \). For our function, it becomes \( f_x = 2x - 3y \).
• Similarly, the partial derivative with respect to \( y \) is \( \frac{\partial f}{\partial y} = -3x + 8y \).

These derivatives give us the slope of the tangent lines to the surface in the \( x \) and \( y \) directions, respectively. Taking partial derivatives is a step towards finding the gradient, which is crucial for exploring directional changes in the function's value.

The directional derivative gives insight into how a function changes in a specific direction from a point. Essentially, it is the rate of change of the function as you move in a specified direction, defined by a vector. For any function \( f(x, y) \), once we have its gradient \( abla f \), we can calculate the directional derivative in the direction of a vector \( \mathbf{u} \).
This is formulated as the dot product of the gradient and the direction vector: \[ D_\mathbf{u}f = abla f \cdot \mathbf{u} \]

• For the given point \( P(1,2) \), the gradient we computed was \( (-4, 13) \).
• The task becomes finding \( \mathbf{u} \) such that \( abla f \cdot \mathbf{u} = 14 \), indicating a specific rate of change.

The directional derivative is significant in optimization and various applications where understanding how changes along different vectors affect the function's outcome is essential.

A unit vector is a vector with a magnitude of 1. It is used to specify directions without considering magnitude. When calculating a directional derivative, a unit vector \( \mathbf{u} \) is essential because it provides a "purified" direction, ensuring that the rate of change measurement is solely attributed to the function's characteristics, not the length of the direction vector.
For directional derivatives:

• We require \( \mathbf{u} \) to satisfy the condition \( \| \mathbf{u} \| = 1 \), meaning \( u_1^2 + u_2^2 = 1 \).
• In this scenario, finding \( \mathbf{u} \) involves solving a system of equations by incorporating the dot product with the gradient.

The use of a unit vector ensures consistent and reliable results when exploring how functions behave in various directions.