Problem 13 - Calculus an Applied Approach

Question

Find the first partial derivatives with respect to \(x\) and with respect to \(y .\) $$ z=\ln \frac{x+y}{x-y} $$

Step-by-Step Solution

Answer

\(\frac{dz}{dx} = \frac{1}{x+y} - \frac{1}{x-y}\) and \(\frac{dz}{dy} = \frac{1}{x+y} + \frac{1}{x-y}\)

Step 1: Rewrite the Function

Begin by rewriting the function to separate the logarithm from the fraction, resulting in \(z = \ln{(x+y)} - \ln{(x-y)} \). This is based on the logarithmic property that the \(\ln (a/b)= \ln a - \ln b.\)

Step 2: Compute the Partial Derivative with Respect to \(x\)

Now, the derivative of the function with respect to \(x\) can be established using chain rule, i.e., \(\frac{dz}{dx} = \frac{d}{dx}\ln{(x+y)} - \frac{d}{dx}\ln{(x-y)}\). This simplifies to \(\frac{dz}{dx} = \frac{1}{x+y} - \frac{1}{x-y}\).

Step 3: Compute the Partial Derivative with Respect to \(y\)

Similarly, the derivative of the function with respect to \(y\) can be calculated as \(\frac{dz}{dy} = \frac{d}{dy}\ln{(x+y)} - \frac{d}{dy}\ln{(x-y)}\). On simplifying, we have \(\frac{dz}{dy} = \frac{1}{x+y} + \frac{1}{x-y}\).

Key Concepts

Logarithmic Differentiation

Logarithmic differentiation is a technique in calculus used to differentiate functions by taking the natural logarithm (ln) of both sides, particularly when dealing with products, quotients, or powers that would otherwise be more complicated to differentiate.

In the given exercise, logarithmic differentiation is applied to reduce the complexity of the derivative of the logarithm of a quotient. By using the property that \(\ln(\frac{a}{b}) = \ln(a) - \ln(b)\), we can simplify the original function \(z=\ln\frac{x+y}{x-y}\) into a difference of two logarithms.

This simplification leads to each term becoming a separate function which is easier to differentiate. The derivatives of \(\ln(x+y)\) and \(\ln(x-y)\) with respect to \(x\) and \(y\) can then be computed separately. The derivative of \(\ln(u)\), where \(u\) is a differentiable function, is \(\frac{1}{u}\frac{du}{dx}\) when differentiating with respect to \(x\), and similar for \(y\).

Understanding logarithmic differentiation helps in simplifying the process of finding derivatives of complex functions and is a necessary tool for working with logarithmic functions.

Chain Rule

The chain rule is a fundamental rule in calculus used for finding the derivative of a composite function. It states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.

In the context of our exercise, the chain rule is utilized when we separate the logarithmic function into two parts, \(z = \ln(x+y)\) and \(z = \ln(x-y)\), and then take the derivative with respect to \(x\) and \(y\). For example, when differentiating \(\ln(x+y)\) with respect to \(x\), we consider \(x+y\) as the inner function. The outer function is the natural logarithm itself.

Therefore, to find \(\frac{d}{dx}\ln(x+y)\), we compute \(\frac{1}{x+y}\) (derivative of the \(\ln\) function), and multiply it by \(\frac{d(x+y)}{dx}\) (derivative of the inner function), which in this case is simply 1. Similar steps are taken when differentiating with respect to \(y\). The chain rule is a versatile tool in differentiating complex functions, especially when dealing with functions of functions, which often occur in higher-level mathematics.

Multivariable Calculus

Multivariable calculus extends the principles of calculus to functions of several variables. Unlike a single-variable function which has a single input and a single output, multivariable functions can have two or more inputs, resulting in a multidimensional output.

In this exercise, the function \(z\) depends on both \(x\) and \(y\), making it a multivariable function. Finding partial derivatives is a key concept within multivariable calculus, as it measures how \(z\) changes as one of the inputs varies, while the other input is held constant.

Partial derivatives with respect to \(x\) and \(y\) have been calculated in the solution. These derivatives help in understanding the slope of the surface \(z\) in the direction of each of the axes in the \(xy\)-plane. They are denoted as \(\frac{\partial z}{\partial x}\) and \(\frac{\partial z}{\partial y}\) respectively. Multivariable calculus is fundamental in fields that involve optimization, contour mapping, and in general, any scenario that requires analysis of functions with more than one variable.