Problem 18 - Calculus Single Variable

Question

Calculate the derivative with respect to \(x\) of the given expression. \(\ln (1 / x)\)

Step-by-Step Solution

Answer

The derivative is \(-\frac{1}{x}\).

Step 1: Identify the Function

First, recognize that you are asked to differentiate the function \( f(x) = \ln \left( \frac{1}{x} \right) \). This is a logarithmic function.

Step 2: Apply the Logarithm Property

Use the property of logarithms that \( \ln \left( \frac{a}{b} \right) = \ln(a) - \ln(b) \) to rewrite the function as \( f(x) = \ln(1) - \ln(x) \). Since \( \ln(1) = 0 \), the function simplifies to \( f(x) = -\ln(x) \).

Step 3: Differentiate the Simplified Function

Now that we have \( f(x) = -\ln(x) \), differentiate it with respect to \( x \). The derivative of \( \ln(x) \) is \( \frac{1}{x} \), so the derivative of \(-\ln(x)\) is \(-\frac{1}{x} \).

Step 4: Write the Final Derivative

Thus, the derivative of \( \ln \left( \frac{1}{x} \right) \) with respect to \( x \) is \( -\frac{1}{x} \).

Key Concepts

Understanding Logarithmic Functions

Logarithmic functions play a critical role in calculus, particularly when dealing with growth processes, radioactive decay, and interest calculations. A logarithmic function is typically written in the form \( \ln(x) \), which refers to the natural logarithm of \( x \). The natural logarithm is the logarithm to the base \( e \), where \( e \) is approximately equal to 2.71828. Understanding the properties of logarithms is essential for manipulation and simplification of expressions. Some key properties include:

• Product property: \( \ln(ab) = \ln(a) + \ln(b) \)
• Quotient property: \( \ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b) \)
• Power property: \( \ln(a^b) = b\ln(a) \)

In the original exercise, we used the quotient property to simplify \( \ln\left(\frac{1}{x}\right) \) to \( \ln(1) - \ln(x) \). Given that \( \ln(1) = 0 \), the expression simplifies to \(-\ln(x)\), making it easier to work with when finding derivatives.

Applying Derivative Rules

Derivative rules are tools in calculus that help find the rate at which a function is changing at any given point. For logarithmic functions, the derivative is derived using specific rules tailored for natural logs. The derivative of the function \( \ln(x) \) is \( \frac{1}{x} \). This rule is straightforward but incredibly powerful when combined with other rules for more complex functions:

• Chain rule: Useful for differentiating composite functions, it states \( \frac{d}{dx}f(g(x)) = f'(g(x)) \, g'(x) \).
• Product rule: \( \frac{d}{dx}[u(x) \cdot v(x)] = u'(x)v(x) + u(x)v'(x) \).
• Quotient rule: Used when dealing with division, it is expressed as \( \frac{d}{dx}\left(\frac{u(x)}{v(x)}\right) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2} \).

In the solution to the exercise, understanding the derivative rule for \( \ln(x) \) as \( \frac{1}{x} \) allowed for direct computation of the derivative of \(-\ln(x)\) to be \(-\frac{1}{x} \). These rules simplify the process of handling complex differential equations.

An Overview of Calculus

Calculus is a branch of mathematics that focuses on change; it is divided into differential calculus and integral calculus. Differentiation, the main focus of this exercise, examines the rate of change of a quantity. It allows us to find the slope of curves and solve problems involving motion, growth, and optimization.In the context of differentiation:

• Function: Described as a relation between a set of inputs and a set of possible outputs where each input is linked to one output.
• Derivative: Represents the rate of change or slope of the curve of a function. It is the fundamental concept in differential calculus.

Differentiation is not confined to algebraic functions, but extends to understand logarithmic, exponential, and trigonometric functions as well. By differentiating \( \ln\left(\frac{1}{x}\right) \), we combined logarithmic property manipulation with differentiation rules to find the solution, illustrating how calculus deeply integrates with various types of functions. Embracing calculus enables problem-solving in a wide range of fields from physics to finance.