Problem 297 - Calculus 3

Question

For the following exercises, find the derivative of the function. \(f(x, y, z)=\ln (x y+y z+z x) \) at point \((-9,-18,-27)\) in the direction the function increases most rapidly

Step-by-Step Solution

Answer

The gradient at \((-9, -18, -27)\) is \(\left(-\frac{5}{99}, -\frac{4}{99}, -\frac{3}{99}\right)\).

Step 1: Find the Gradient of the Function

The direction in which a function increases most rapidly is given by the gradient of the function. For a function \( f(x, y, z) \), the gradient \( abla f \) is a vector of partial derivatives: \( abla f(x, y, z) = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right) \). First, compute each partial derivative using the chain rule. Start with1. \( \frac{\partial f}{\partial x} = \frac{1}{x y + y z + z x} \cdot (y + z) \),2. \( \frac{\partial f}{\partial y} = \frac{1}{x y + y z + z x} \cdot (x + z) \),3. \( \frac{\partial f}{\partial z} = \frac{1}{x y + y z + z x} \cdot (x + y) \).

Step 2: Evaluate the Partial Derivatives at the Given Point

Now plug the point \((-9, -18, -27)\) into the partial derivatives:1. \( \frac{\partial f}{\partial x}(-9, -18, -27) = \frac{1}{(-9)(-18) + (-18)(-27) + (-27)(-9)} \cdot (-18 - 27) \),2. \( \frac{\partial f}{\partial y}(-9, -18, -27) = \frac{1}{(-9)(-18) + (-18)(-27) + (-27)(-9)} \cdot (-9 - 27) \),3. \( \frac{\partial f}{\partial z}(-9, -18, -27) = \frac{1}{(-9)(-18) + (-18)(-27) + (-27)(-9)} \cdot (-9 - 18) \).

Step 3: Simplify and Compute the Gradient Vector

Calculate the denominator: \((-9)(-18) + (-18)(-27) + (-27)(-9) = 162 + 486 + 243 = 891\).Evaluate the numerators and find the partial derivatives:1. \( \frac{\partial f}{\partial x}(-9, -18, -27) = \frac{-45}{891} = -\frac{5}{99} \),2. \( \frac{\partial f}{\partial y}(-9, -18, -27) = \frac{-36}{891} = -\frac{4}{99} \),3. \( \frac{\partial f}{\partial z}(-9, -18, -27) = \frac{-27}{891} = -\frac{3}{99} \).Therefore, the gradient vector \( abla f(-9, -18, -27) = \left(-\frac{5}{99}, -\frac{4}{99}, -\frac{3}{99}\right) \).

Step 4: Determine the Direction of Maximum Increase

The direction of the maximum increase of the function is given by the gradient vector \( abla f \). Hence, the direction where \( f(x, y, z) \) increases most rapidly at \((-9, -18, -27)\) is the vector \( \left(-\frac{5}{99}, -\frac{4}{99}, -\frac{3}{99}\right) \).

Key Concepts

Partial Derivatives

In multivariable calculus, partial derivatives are used to investigate how a function changes when only one variable is altered, keeping the others constant. This is quite similar to ordinary derivatives in single-variable calculus but applied to functions of multiple variables.
For a function \( f(x, y, z) \), its partial derivatives are represented as \( \frac{\partial f}{\partial x} \), \( \frac{\partial f}{\partial y} \), and \( \frac{\partial f}{\partial z} \). These derivatives measure the rate of change of the function with respect to each variable.
Here are the steps to compute them:

• Hold all other variables constant.
• Differentiate the function with respect to the variable of interest.

For the function \( f(x, y, z)=\ln (x y+y z+z x) \), this means focusing on each variable individually while calculating the derivative.

Directional Derivative

The directional derivative of a multivariable function represents the rate at which the function changes as one moves in a specific direction from a given point. It provides insight into how the function behaves in different directions.
To find the directional derivative, one can use the gradient vector \( abla f \), which effectively points in the direction of the greatest rate of increase of the function. The gradient is essentially a combination of partial derivatives.
To calculate the directional derivative in a specific direction, take the dot product of the gradient vector with a unit vector in the desired direction. The unit vector must have a magnitude of 1, ensuring it only reflects direction and not magnitude.
This process highlights the power of directional derivatives in understanding how a function evolves, especially when seeking maxima or minima.

Chain Rule

The chain rule is a crucial technique in calculus for handling the differentiation of composite functions. It connects the derivative of a composition of functions to the derivatives of its components.
When dealing with multivariable functions, the chain rule allows you to compute derivatives when variables are interdependent or when the function is not easily separable.
For instance, if you have a function \( f(u) \), where \( u = g(x, y, z) \), the derivative of \( f \) with respect to \( x \) involves not only the derivative of \( f \) but also the derivative of \( g \). It becomes \( \frac{df}{dx} = \frac{df}{du} \cdot \frac{du}{dx} \).
The application of the chain rule is critical in finding the partial derivatives successfully, ensuring an accurate portrayal of how variables influence each other in a complex system.