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The gradient vector of a function is a crucial concept in multivariable calculus. It represents the direction and rate of the steepest ascent of a scalar field. For a function of two variables, like our function, the gradient is a two-dimensional vector containing the partial derivatives with respect to each variable.

• The gradient of a function \(f(x, y)\) is denoted as \(abla f(x, y)\).
• The components include \(\frac{\partial f}{\partial x}\) and \(\frac{\partial f}{\partial y}\), representing the rate of change of the function in the directions of the x-axis and y-axis, respectively.

For our exercise, the function is \(f(x, y) = \cos(3x-y)\), yielding the gradient \(abla f(x, y) = \langle -3\sin(3x-y), \sin(3x-y) \rangle\). This indicates how quickly and in which direction the function's value is changing most rapidly.
Evaluating at point \(P(\pi/6, \pi/4)\) gives us the vector \(\langle -3, 1 \rangle\), showing that the function's steepest ascent at this point is in this direction.

The directional derivative is a fundamental concept when analyzing changes in functions. It generalizes the concept of a derivative from one-dimensional functions to functions of multiple variables. Essentially, it measures how a function changes as you move in a specific direction.

• The directional derivative of a function \(f\) at a point \(P\) in the direction of a vector \(\mathbf{v}\) is given by the dot product of the gradient \(abla f(P)\) and the unit vector \(\mathbf{u}\) in the direction of \(\mathbf{v}\).
• If \(\mathbf{v}\) is not a unit vector, you first normalize it, dividing it by its magnitude to get \(\mathbf{u}\).

In the context of our problem, the direction we're interested in is where the function decreases most, which is in the opposite direction of the gradient. By calculating this, we determine that the directional derivative, representing the rate of change in the negative gradient direction, equals \(-\sqrt{10}\), confirming the rapid reduction of the function's value at that point.

Partial derivatives are the building blocks for understanding gradients and directional derivatives. When dealing with functions of several variables, a partial derivative represents how the function changes as you vary one variable while keeping others constant.

• The partial derivative with respect to \(x\), denoted \(\frac{\partial f}{\partial x}\), considers changes only along the x-axis.
• Similarly, \(\frac{\partial f}{\partial y}\) examines changes along the y-axis.

In our example, the function \(f(x, y) = \cos(3x - y)\), has partial derivatives computed as:\[ \frac{\partial f}{\partial x} = -3\sin(3x - y)\] and \[ \frac{\partial f}{\partial y} = \sin(3x - y)\]. These partial derivatives are crucial components of the gradient, which we evaluated at a specific point to understand the direction and rate of maximum function decrease. Understanding the role of partial derivatives allows us to appreciate how individual factors contribute to the overall gradient of the function.