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Chapter 12: Problem 16

Find the first partial derivatives of the following functions. $$f(x, y)=\ln (x / y)$$

Short Answer

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Question: Find the first partial derivatives of the function $$f(x, y) = \ln(x / y)$$. Answer: The first partial derivatives of the function $$f(x, y) = \ln(x / y)$$ are: $$\frac{\partial f}{\partial x} = \frac{1}{x}$$ $$\frac{\partial f}{\partial y} = -\frac{1}{y}$$

Step by step solution

01

Find the partial derivative with respect to x

To find the first partial derivative with respect to x, we will differentiate the function f(x, y) with respect to x and treat y as a constant. The partial derivative of a function with respect to x is denoted as $$\frac{\partial f}{\partial x}$$. First, we rewrite the function as: $$f(x, y) = \ln(x) - \ln(y)$$ Now, for differentiation: $$\frac{\partial f}{\partial x} = \frac{\partial}{\partial x}(\ln(x) - \ln(y))$$ Since the derivative of $$\ln(x)$$ with respect to x is $$\frac{1}{x}$$ and the derivative of $$\ln(y)$$ with respect to x is 0, we get: $$\frac{\partial f}{\partial x} = \frac{1}{x}$$
02

Find the partial derivative with respect to y

Similarly, to find the first partial derivative with respect to y, we will differentiate the function f(x, y) with respect to y and treat x as a constant. The partial derivative of a function with respect to y is denoted as $$\frac{\partial f}{\partial y}$$. For differentiation: $$\frac{\partial f}{\partial y} = \frac{\partial}{\partial y}(\ln(x) - \ln(y))$$ Since the derivative of $$\ln(y)$$ with respect to y is $$-\frac{1}{y}$$ and the derivative of $$\ln(x)$$ with respect to y is 0, we get: $$\frac{\partial f}{\partial y} = -\frac{1}{y}$$
03

Write the final answer

The first partial derivatives of the function $$f(x, y) = \ln(x / y)$$ are: $$\frac{\partial f}{\partial x} = \frac{1}{x}$$ $$\frac{\partial f}{\partial y} = -\frac{1}{y}$$

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Multivariable Calculus
In multivariable calculus, we extend the concepts from single-variable calculus to functions of several variables. One key aspect of this field is the study of how these functions change in response to changes in each of their variables. When dealing with functions like \( f(x, y) = ln(x/y) \), we are interested in understanding how a small change in \(x\) or \(y\) affects the value of \(f\).

When computing these changes, we make use of partial derivatives, which are similar to the derivatives from single-variable calculus, but they focus on the change in one variable while all other variables are held constant. This provides us with valuable information on the function's behavior with respect to each variable individually. Calculating partial derivatives is a fundamental skill in fields like physics, engineering, economics, and beyond, where functions often depend on multiple, interrelated variables.
Derivative of Natural Logarithm
The natural logarithm function, often denoted as \(ln(x)\), is an important function in mathematics with a unique property: its derivative with respect to \(x\) is \(1/x\). This derivative is significant due to the natural logarithm's role in many areas including growth processes, compound interest, and in calculus itself where it's used in integration and differentiation.

Understanding how \(ln(x)\) behaves when the variable changes is crucial in the study of calculus. When performing differentiation, the simple form of its derivative makes it easier to calculate the rates of change in more complex functions involving \(ln\). As seen in the provided solution, recognizing that the derivative of \(ln(y)\) is zero when differentiating with respect to \(x\) is essential, because \(y\) is treated as a constant in that case, leading to simplifications in the calculation of partial derivatives.
First Partial Derivative
The concept of the first partial derivative is a stepping stone to understanding how multivariable functions change in one direction at a time. It's the equivalent to the derivative of single-variable functions, but specifically targets one variable while holding others constant. For instance, \( \frac{\partial f}{\partial x} \) tells us the rate at which \(f(x, y)\) changes as only \(x\) varies, with \(y\) remaining fixed.

To calculate the first partial derivative, we apply the rules of differentiation known from single-variable calculus, but we treat all other variables as constants. This can sometimes simplify the process, as we ignore the changes along other variable axes. Understanding first partial derivatives is essential for predicting the behavior of more complex systems and forms the basis for concepts like directional derivatives and gradients, expanding the analysis tools for multivariable functions.

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