91Ó°ÊÓ

Understanding partial derivatives is a crucial part of working with functions of multiple variables. When we talk about partial derivatives, we are finding how a function changes as one variable changes while keeping the others constant.

For example, consider the function \(f(x, y)=\frac{(x-y)}{(x+y)}\), which depends on both \(x\) and \(y\). To find its behavior in each direction, we calculate its partial derivatives:

• The partial derivative with respect to \(x\) shows how \(f\) changes as \(x\) changes and \(y\) remains fixed.
• Likewise, the partial derivative with respect to \(y\) indicates how \(f\) shifts with adjustments in \(y\) while \(x\) stays the same.

For this function, the partial derivatives are:

• \(\frac{\partial f}{\partial x} = \frac{2y}{(x+y)^2}\)
• \(\frac{\partial f}{\partial y} = \frac{-2x}{(x+y)^2}\)

These calculations help pinpoint the rate and direction of change at any point \((x, y)\).

The gradient vector has a special place in calculus, especially when dealing with directional derivatives. After computing partial derivatives, they are combined into the gradient vector.

• It is essentially a vector that points in the direction of steepest ascent on a surface represented by a function.
• Mathematically, the gradient vector of a function \(f(x, y)\) is denoted as \(abla f\). For our function, \(abla f = (\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y})\).

At the specific point \((-\frac{1}{2}, \frac{3}{2})\), the earlier calculated partial derivatives give us the gradient vector: \[abla f\left(-\frac{1}{2}, \frac{3}{2}\right) = (3, 1).\] This vector, \((3, 1)\), indicates the direction where the function increases the most at that point, while the opposite direction indicates the steepest descent. The length of the gradient vector also tells us how quickly the function increases in its steepest direction.

A unit vector is a vector that has a length of one. It is crucial in various vector operations, especially when dealing with directional derivatives. The idea of a unit vector comes into play when you want to specify a direction without altering magnitude when multiplying by another vector.

For finding directional derivatives, ensure that the direction is specified by a unit vector. If you have any vector \(\mathbf{v} = (a, b)\), the corresponding unit vector \(\mathbf{u}\) is calculated as:\[ \mathbf{u} = \frac{(a, b)}{\sqrt{a^2 + b^2}}. \]

• This process normalizes the vector, adjusting it to have a length of one.
• Using the gradient \((3, 1)\), its unit vector is \(\left(\frac{3}{\sqrt{10}}, \frac{1}{\sqrt{10}}\right)\), showing that this unit vector keeps the balance of the original vector's direction. It succinctly points in the same way as the original but on a unit scale.

Unit vectors provide a standard, consistent way to describe direction, making calculations like the directional derivative cleaner and more interpretable.