Problem 24 - Calculus with Concepts in Calculus

Question

Find \(\partial w / \partial u\) and \(\partial w / \partial v\). $$ w=e^{x / y}+e^{z / x} ; x=\frac{\ln u}{v}, y=\ln u, z=\frac{\ln u}{u v} $$

Step-by-Step Solution

Answer

Use the chain rule to find both derivatives, combining the partials using the expressions for \(w, x, y, z\).

Step 1: Chain Rule Overview

We need to find the partial derivatives \(\frac{\partial w}{\partial u}\) and \(\frac{\partial w}{\partial v}\) using the chain rule, since \(w\) is a function of \(x, y, z\), which are in turn functions of \(u\) and \(v\).

Step 2: Express Partial Derivatives

The chain rule for multiple variables gives:\[\frac{\partial w}{\partial u} = \frac{\partial w}{\partial x} \cdot \frac{\partial x}{\partial u} + \frac{\partial w}{\partial y} \cdot \frac{\partial y}{\partial u} + \frac{\partial w}{\partial z} \cdot \frac{\partial z}{\partial u} \]and \[\frac{\partial w}{\partial v} = \frac{\partial w}{\partial x} \cdot \frac{\partial x}{\partial v} + \frac{\partial w}{\partial y} \cdot \frac{\partial y}{\partial v} + \frac{\partial w}{\partial z} \cdot \frac{\partial z}{\partial v} \].

Step 3: Compute \(\partial w/\partial x\), \(\partial w/\partial y\), and \(\partial w/\partial z\)

Calculate the partial derivatives of \(w\) with respect to \(x, y, z\):\[\frac{\partial w}{\partial x} = \frac{1}{y} e^{x / y} - \frac{z}{x^2} e^{z / x},\frac{\partial w}{\partial y} = -\frac{x}{y^2} e^{x / y},\frac{\partial w}{\partial z} = \frac{1}{x} e^{z / x}.\]

Step 4: Compute \(\partial x/\partial u\) and \(\partial x/\partial v\)

Find the derivatives for \(x = \frac{\ln u}{v}\):\[\frac{\partial x}{\partial u} = \frac{1}{uv}, \quad\text{and}\quad \frac{\partial x}{\partial v} = -\frac{\ln u}{v^2}.\]

Step 5: Compute \(\partial y/\partial u\) and \(\partial y/\partial v\)

For \(y = \ln u\):\[\frac{\partial y}{\partial u} = \frac{1}{u}, \quad\text{and}\quad \frac{\partial y}{\partial v} = 0.\]

Step 6: Compute \(\partial z/\partial u\) and \(\partial z/\partial v\)

For \(z = \frac{\ln u}{uv}\):\[\frac{\partial z}{\partial u} = \frac{1 - \ln u}{u^2v}, \quad\text{and}\quad \frac{\partial z}{\partial v} = -\frac{\ln u}{uv^2}.\]

Step 7: Combine Results for \(\partial w/\partial u\)

Substitute all partial derivatives into the chain rule expression for \(\frac{\partial w}{\partial u}\) and simplify to obtain the final expression.

Step 8: Combine Results for \(\partial w/\partial v\)

Substitute all partial derivatives into the chain rule expression for \(\frac{\partial w}{\partial v}\) and simplify to obtain the final expression.

Key Concepts

Partial Derivatives

Partial derivatives are essential when dealing with functions of multiple variables since they allow us to focus on how a function changes with respect to one variable at a time, while keeping the others constant. This concept is pivotal in calculus and finds applications across physics, engineering, and economics. In the original exercise, we aimed to determine \(\frac{\partial w}{\partial u}\) and \(\frac{\partial w}{\partial v}\), where \(w\) is influenced by three intermediate variables \(x, y, z\).

• Chain Rule: The chain rule for partial derivatives acknowledges that the changes in \(w\) are indirectly driven by changes in \(u\) and \(v\) through \(x, y, z\).
• We express \(\frac{\partial w}{\partial u}\) using the derivatives of \(w\) with respect to \(x, y, z\), multiplying each by the respective derivative of \(x, y, z\) concerning \(u\). The same applies when computing \(\frac{\partial w}{\partial v}\).

The chain rule is a vital concept in differential calculus, allowing us to understand and compute the rate of change in multidimensional scenarios.

Functions of Multiple Variables

Functions of multiple variables are mathematical expressions where the output depends on several inputs. These functions are prevalent in calculations involving spatial dynamics or any system with interdependent variables. Let's break it down with an example from the original problem:

• Interdependency: The function \(w = e^{x/y} + e^{z/x}\) has \(x\), \(y\), and \(z\) as variables, each depending on further variables \(u\) and \(v\). This nested dependency structure is typical in multivariable functions.
• The solution involves expressing these complex interdependencies via partial derivatives to manage the change concerning a specific variable.

Recognizing these relationships and understanding how the dependent variables change in unison is key to mastering functions of multiple variables. It allows the mapping of complex systems into more manageable mathematical frames.

Exponential Functions

Exponential functions are crucial in both natural sciences and economics due to their ability to model growth processes and decay phenomena. They become even more interesting in multivariable contexts, where they can be combined with other functions, as in the exercise.

• Understanding Exponents: In the function \(w = e^{x/y} + e^{z/x}\), both parts are exponential functions, which have a constant base \(e\) (Euler's number), and variable exponents. This structure signifies exponential growth or decay concerning one or multiple factors.
• Partial Derivatives of Exponentials: When deriving partial derivatives of exponential functions, as seen in the solution, it is vital to apply the chain rule accurately, ensuring the effect of the exponent's rate of change is precisely accounted for.

The beauty of exponential functions lies in their mathematical simplicity opposed to their powerful ability to model real-world dynamics effectively. Their representation in scenarios involving multiple variables showcases their flexibility and utility.