91Ó°ÊÓ

The gradient vector is an essential concept when working with multivariable functions. Think of it as a compass that points in the direction where the function increases at its fastest rate. For a function written in terms of two variables, let's say \( f(x, y) \), the gradient is a vector composed of its partial derivatives in respect to \( x \) and \( y \).
The notation for the gradient vector is \( abla f(x, y) \) and specifically it's represented as \( \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right) \). In essence, each component of the gradient vector tells us how much the function changes along each axis.
In our problem, we calculated the gradient of the function \( f(x, y) = \sin(xy) \). At the point \((1,0)\), the gradient vector turned out to be \( (0, 1) \). This means the function increases its fastest along the direction defined by \( (0, 1) \), or more simply along the positive \( y \)-axis.

Partial derivatives help us understand how a multivariable function behaves when we tweak one variable while keeping others constant. They represent the function's rate of change with respect to one variable at a time, creating a crucial building block for the gradient vector.
For instance, while working with the function \( f(x, y) = \sin(xy) \), we compute the partial derivative with respect to \( x \) as \( \frac{\partial f}{\partial x} = y \cos(xy) \) and with respect to \( y \) as \( \frac{\partial f}{\partial y} = x \cos(xy) \).
Evaluating these at \( (1,0) \), we found that \( \frac{\partial f}{\partial x} = 0 \) and \( \frac{\partial f}{\partial y} = 1 \). Each of these values gives us a snapshot of how the function changes along the \( x \)- and \( y \)-axes, respectively, at that particular point.

The directional derivative gives us a valuable insight into how a function changes at a point in a given direction. It's like asking, "How steep would the hill be if I decided to walk in this particular direction?"
Mathematically, the directional derivative of a function \( f \) at a point \( (x_0, y_0) \) in the direction given by a unit vector \( \mathbf{u} = (u_1, u_2) \) is computed using the dot product: \( D_{\mathbf{u}} f(x_0, y_0) = abla f(x_0, y_0) \cdot \mathbf{u} \).
For our exercise, though we specifically focused on the gradient, the concept is related. The maximum directional derivative, and thus the maximum rate of change of the function, occurs in the gradient's direction. At the point \((1,0)\), this slope is exactly 1, showing the steepness of the climb if you head in the \( (0, 1) \) or positive \( y \)-direction.