91Ó°ÊÓ

Partial derivatives are crucial for understanding how a function changes with respect to one variable while holding the others constant. In the context of the function \(f(p, q, r) = \arctan(p q r)\), we find each partial derivative by differentiating with respect to \(p\), \(q\), and \(r\).
This process allows us to understand how changes in \(p\), \(q\), or \(r\) influence the entire function. Breaking it down:

• For \(\frac{\partial f}{\partial p}\), treat \(q\) and \(r\) as constants, yielding \(\frac{q r}{1+(p q r)^2}\).
• For \(\frac{\partial f}{\partial q}\), keep \(p\) and \(r\) constant, resulting in \(\frac{p r}{1+(p q r)^2}\).
• For \(\frac{\partial f}{\partial r}\), treat \(p\) and \(q\) as constants, resulting in \(\frac{p q}{1+(p q r)^2}\).

These derivatives form the gradient vector, which is central to understanding the rate of change of the function.

The rate of change of a function at a specific point gives us insight into the behavior of that function in various directions. Imagine moving slightly in any direction from a point; the rate of change tells us how the function's value will change.
In our specific problem, we are looking for the maximum rate of change of the function \(f(p, q, r)\) at the point \((1, 2, 1)\).
The steps to follow:

• First, we compute the gradient vector, which combines the partial derivatives to show how the function changes in each variable direction.
• The maximum rate of change is determined by the magnitude of this gradient vector.

Thus, knowing the rate of change at any point helps predict how the function will behave in that vicinity.

Directional derivatives extend the concept of partial derivatives by measuring how a function changes in a specific direction. To find a directional derivative, we use a unit vector in the desired direction and the gradient vector.
The directional derivative along a unit vector \(\mathbf{u}\) is given by the dot product:
\(D_\mathbf{u}f = abla f \cdot \mathbf{u}\)
Where \(abla f\) is the gradient vector. This measure gives insight on how steeply the function rises or falls in that direction.
In scenarios like ours, the gradient vector itself provides a specific direction with the steepest ascent, thus efficiently calculating the rate at which the function changes in any direction.

The magnitude of the gradient vector reflects the maximum rate of change of a function at a particular point.
This is because the gradient not only points in the direction of greatest increase, but its length (magnitude) also tells us how steep that rise is.
To find the magnitude, apply the Pythagorean theorem to the gradient vector components:
For our gradient at point \((1, 2, 1)\), \(abla f(1, 2, 1) = \left(\frac{2}{5}, \frac{1}{5}, \frac{2}{5}\right)\).
The magnitude is \[\sqrt{\left(\frac{2}{5}\right)^2 + \left(\frac{1}{5}\right)^2 + \left(\frac{2}{5}\right)^2} = \frac{3}{5}\]
Thus, \(\frac{3}{5}\) is the maximum rate of increase for \(f\) at the point \((1, 2, 1) \).
Understanding both direction and magnitude provides a complete picture of how functions change spatially.