Problem 41 - Calculus of a Single Variable

Question

In Exercises 41–64, find the derivative of the function. $$ f(x)=\ln (3 x) $$

Step-by-Step Solution

Answer

The derivative of \( f(x) = \ln(3x) \) is \( f'(x) = 1/x \).

Step 1: Identify the Outer and Inner Function

Firstly, the function \(f(x) = \ln(3x)\) is a composition of two functions, namely \(u(x) = \ln(x)\) and \(v(x) = 3x\). In this equation, \(u(x)\) is considered as the 'outer function' and \(v(x)\) is considered as the 'inner function'.

Step 2: Apply the Chain Rule

The Chain Rule states that the derivative of a composition of functions is the derivative of the outside function evaluated at the inside function, times the derivative of the inside function. Mathematically, it's expressed as \(f'(x) = u'(v(x)) \cdot v'(x)\).

Step 3: Evaluate the Derivatives of the Inner and Outer Function

The derivative of the outer function \(u(x) = \ln(x)\) is \(u'(x) = 1/x\) and the derivative of the inside function \(v(x) = 3x\) is \(v'(x) = 3\).

Step 4: Substitute the Derivatives into the Chain Rule Formula

Replace \(u'(v(x))\) and \(v'(x)\) in the formula with their respective results. This gives us, \(f'(x) = 1/(3x) \cdot 3\).

Step 5: Simplify the Result

Now, simplify the equation to arrive at the final solution. The 3 in the denominator and the 3 in the numerator cancel out, resulting in \(f'(x) = 1/x\).

Key Concepts

Differentiation of Logarithmic Functions

In calculus, the differentiation of logarithmic functions is a fundamental concept that allows us to compute the rate at which a logarithmic function changes. A logarithmic function, typically of the form \( \ln(x) \), expresses the logarithm of a variable. The derivative of this function is uniquely described by the reciprocal of the variable. In other words, if \( f(x) = \ln(x) \) then, according to the rules of differentiation, \( f'(x) = \frac{1}{x} \).

Understanding how to differentiate \( \ln \) functions becomes especially vital when these functions are part of a more complex function composition, as we often encounter logarithms in natural sciences and economics to describe growth and decay processes. It is the simplicity of its derivative that often simplifies the overall differentiation process. For a natural logarithm of a product, for instance, \( \ln(ax) \) the differentiation involves the application of additional rules such as the Chain Rule.

Applying the Chain Rule

The Chain Rule is a quintessential tool in calculus for finding the derivative of composed functions. When a function \( y \) is a combination of two functions, say \( u \) and \( v \) where \( u \) depends on \( v \) which in turn depends on \( x \), the Chain Rule provides a method to differentiate \( y \) with respect to \( x \).

The Chain Rule formula states that \( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dv} \cdot \frac{dv}{dx} \). It allows you to compute derivatives step by step, by taking the derivative of the outer function (as though the inner function were a simple variable) and then multiplying by the derivative of the inner function. In practice, this can be visualized as 'peeling an onion,' differentiating layer by layer. Correctly identifying the inner and outer functions is crucial to effectively apply the Chain Rule. This rule facilitates the differentiation of more complex structures by breaking them down into simpler parts.

Derivatives in Calculus

Derivatives are one of the cornerstone concepts of calculus, representing an essential mathematical tool used to examine the rate at which quantities change. When we talk about a derivative, we are essentially looking for the 'slope' or the rate of change of a function at any given point.

The derivative of a function \( f(x) \) at a point \( x \) is given by \( f'(x) \) and can be interpreted as the instant rate of change of the function with respect to \( x \). It serves as the foundation for many applications across physics, engineering, economics, and beyond. The derivatives not only help in finding rates but are also pivotal in optimization problems, where one seeks to find maximum or minimum values. Mastery of differentiation techniques, including the power rule, product rule, quotient rule, and Chain Rule is essential for any student delving into the world of calculus.

Composition of Functions

The concept of composition of functions is analogous to the fusion of two distinct processes into a single process. In mathematics, when you have two functions, \( f(x) \) and \( g(x) \) the composition of the two, denoted as \( f(g(x)) \), literally means plugging the output of \( g(x) \) into the input of \( f(x) \).

This operation creates a new function that maps inputs from \( g's \) domain to outputs of \( f \) in a chained manner. Composition effects become particularly interesting when the composed functions have distinct properties e.g., when one is a polynomial and the other is a trigonometric function. The ability to differentiate such composed functions using the Chain Rule is a critical skill in calculus, allowing us to understand and describe complex systems in a dynamic way. It's the idea of functional layering that makes composition such a versatile and powerful mathematical tool.