Problem 298 - Calculus 3

Question

For the following exercises, find the derivative of the function. \(f(x, y, z)=\frac{x}{y}+\frac{y}{z}+\frac{z}{x}\) at point \((5,-5,5)\) in the direction the function increases most rapidly

Step-by-Step Solution

Answer

The direction of most rapid increase is \((-\frac{2}{5}, \frac{1}{5}, \frac{2}{5})\).

Step 1: Understand the Gradient Vector

The gradient vector, \(abla f(x, y, z)\), represents the direction of the steepest ascent for the function, which is the direction in which the function increases most rapidly. It consists of the partial derivatives of the function with respect to each variable: \(abla f(x, y, z) = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right)\).

Step 2: Compute the Partial Derivative with Respect to x

Find the partial derivative of \(f(x, y, z)\) with respect to \(x\):\[ \frac{\partial f}{\partial x} = \frac{\partial}{\partial x} \left( \frac{x}{y} + \frac{y}{z} + \frac{z}{x} \right) = \frac{1}{y} - \frac{z}{x^2} \]

Step 3: Compute the Partial Derivative with Respect to y

Find the partial derivative of \(f(x, y, z)\) with respect to \(y\):\[ \frac{\partial f}{\partial y} = \frac{\partial}{\partial y} \left( \frac{x}{y} + \frac{y}{z} + \frac{z}{x} \right) = -\frac{x}{y^2} + \frac{1}{z} \]

Step 4: Compute the Partial Derivative with Respect to z

Find the partial derivative of \(f(x, y, z)\) with respect to \(z\):\[ \frac{\partial f}{\partial z} = \frac{\partial}{\partial z} \left( \frac{x}{y} + \frac{y}{z} + \frac{z}{x} \right) = -\frac{y}{z^2} + \frac{1}{x} \]

Step 5: Evaluate the Gradient Vector at (5, -5, 5)

Substitute the point \((5, -5, 5)\) into the gradient vector \(abla f(x, y, z) = \left( \frac{1}{y} - \frac{z}{x^2}, -\frac{x}{y^2} + \frac{1}{z}, -\frac{y}{z^2} + \frac{1}{x} \right)\): \[ abla f(5, -5, 5) = \left( -\frac{1}{5} - \frac{5}{25}, -\frac{5}{25} + \frac{1}{5}, \frac{5}{25} + \frac{1}{5} \right) \]Simplify each component:\[ = \left( -\frac{1}{5} - \frac{1}{5}, \frac{0}{5} + \frac{1}{5}, \frac{1}{5} + \frac{1}{5} \right) \]\[ = \left( -\frac{2}{5}, \frac{1}{5}, \frac{2}{5} \right) \]

Step 6: Conclusion of Direction of Most Rapid Increase

The direction in which the function increases most rapidly at the point \((5, -5, 5)\) is given by the gradient vector \(abla f(5, -5, 5) = \left( -\frac{2}{5}, \frac{1}{5}, \frac{2}{5} \right)\).

Key Concepts

Partial Derivative

A partial derivative represents how a function changes concerning one of its variables, keeping other variables constant. Thus, it measures the rate of change of the function in that particular direction.
When dealing with functions of several variables, such as the function in the original exercise,

• \( f(x, y, z) = \frac{x}{y} + \frac{y}{z} + \frac{z}{x} \)

the aim is to find out how this function changes if we slightly modify one variable at a time while freezing the others.
Partial derivatives are calculated as follows:

• For \( x \): \( \frac{\partial f}{\partial x} = \frac{1}{y} - \frac{z}{x^2} \)
• For \( y \): \( \frac{\partial f}{\partial y} = -\frac{x}{y^2} + \frac{1}{z} \)
• For \( z \): \( \frac{\partial f}{\partial z} = -\frac{y}{z^2} + \frac{1}{x} \)

These partial derivatives are the components of the gradient vector, which we will discuss next.

Directional Derivative

The directional derivative of a function at a given point tells us how fast the function changes in a specific direction from that point.
Unlike partial derivatives, which evaluate change along the axes, the directional derivative gives a more refined perception, capturing change along any line through the point.
To compute it, we need the gradient vector, \( abla f \), and a direction vector, \( \mathbf{u} \), usually a unit vector:\[ D_\mathbf{u} f(x, y, z) = abla f(x, y, z) \cdot \mathbf{u} \]This expression calculates the dot product between the gradient vector and the unit direction vector. The result tells us the rate of change of the function along that line.
By choosing different direction vectors, one can assess the behavior of the function in various directions.

Steepest Ascent

Steepest ascent refers to the direction in which a function increases the most rapidly at a point. The gradient vector, \( abla f \), inherently points in the direction of the steepest ascent.
This property makes the gradient vector a crucial tool in optimization and analysis of multivariable functions.
In our example, the gradient vector was evaluated as \( \left( -\frac{2}{5}, \frac{1}{5}, \frac{2}{5} \right) \) at point \((5, -5, 5)\). This indicates the direction where the function \( f(x, y, z) = \frac{x}{y} + \frac{y}{z} + \frac{z}{x} \) climbs fastest.
Following this direction reveals not just a path of increase but the most efficient ascent, making it crucial for gradient ascent algorithms used in machine learning and other computational fields.