Problem 63 - Calculus and its applications

Question

Differentiate. $$ f(x)=\ln (9 x) $$

Step-by-Step Solution

Answer

The derivative of \( f(x) = \ln(9x) \) is \( \frac{1}{x} \).

Step 1: Identify the function components

The function given is \( f(x) = \ln (9x) \). It is a logarithmic function, specifically the natural logarithm of a product, \(9x\). The differentiation rule that applies here is the derivative of \( \ln(u) = \frac{1}{u} \cdot \frac{du}{dx} \).

Step 2: Differentiate the outer function

Apply the derivative rule for natural logarithms. The derivative of \( \ln(u) \) is \( \frac{1}{u} \). For our function, \( u = 9x \), so the derivative is \( \frac{1}{9x} \).

Step 3: Differentiate the inner function

Differentiate the inner function \( u = 9x \) with respect to \( x \). The derivative \( \frac{d}{dx}(9x) = 9 \).

Step 4: Apply the chain rule

Combine the results from Step 2 and Step 3 using the chain rule. Multiply the derivative of the outer function by the derivative of the inner function: \( \frac{1}{9x} \cdot 9 = \frac{9}{9x} \).

Step 5: Simplify the expression

Simplify \( \frac{9}{9x} \) to obtain the final result. This simplifies to \( \frac{1}{x} \). Thus, the derivative of \( f(x) = \ln(9x) \) is \( \frac{1}{x} \).

Key Concepts

Logarithmic Functions

The function we are working with, \( f(x) = \ln (9x) \), is a type of logarithmic function. These functions are the inverse of exponential functions and are written in the form \( \log_b(x) \) or \( \ln(x) \) when dealing with the natural logarithm, which uses the base \( e \). In our function, the operation of logarithms simplifies complex multiplication or division problems involving exponentials into a more manageable form. Here's what you need to know about them:- **Properties of Logarithms**: They convert multiplication into addition, division into subtraction, exponentiation into multiplication, and root extraction into division.- **Common Logarithm vs. Natural Logarithm**: The natural logarithm utilizes base \( e \), which is approximately 2.718. This makes it particularly useful in calculus due to its derivative properties.- **Applications**: They are commonly employed in mathematical modeling of growth and decay processes, such as in finance for compound interest calculations, and in physics for measuring decibel levels or radioactive decay. Understanding logarithmic functions is crucial, as they allow us to differentiate complex multiplicative relationships in more straightforward linear terms, exemplified in our function \( \ln (9x) \).

Chain Rule

The chain rule is an essential tool in calculus for differentiating composite functions—those made by combining two or more functions. Think of it like peeling an onion, one layer at a time, because you have an outer function that wraps around an inner function. Here's a breakdown of the chain rule:- **Definition**: If you have a composite function \( f(g(x)) \), the derivative is \( f'(g(x)) \cdot g'(x) \).- **How it Works**: It requires taking the derivative of the outer function first, leaving the inner function intact, and then multiplying it by the derivative of the inner function itself.- **Example in our case**: For \( f(x) = \ln(9x) \), the inner function is \( g(x) = 9x \) and the outer function is \( f(u) = \ln(u) \). First, we differentiate \( \ln(u) \) to get \( \frac{1}{u} \) where \( u = 9x \). Then, the derivative of \( 9x \) is \( 9 \). Multiplying these results gives us \( \frac{1}{9x} \cdot 9 = \frac{1}{x} \).The chain rule helps simplify these components by breaking down the composite function into more digestible parts, making differentiation straightforward.

Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is a special kind of logarithm with the base \( e \), where \( e \approx 2.718 \.et c \). It is particularly useful in calculus due to its simple and elegant derivative properties. Let's explore some key aspects:- **Base \( e \)**: The natural logarithm's base \( e \) makes it unique and invaluable in continuous growth processes because of its mathematical properties.- **Derivative of the Natural Log**: The derivative of \( \ln(x) \) is simply \( \frac{1}{x} \). This function’s simplicity stems from how it behaves with respect to \( e \), making it a staple in calculus.- **Why Use the Natural Logarithm**: Its properties result in simple expressions during differentiation, as seen in our problem's conclusion. Differentiating \( \ln(9x) \) using the rules of logarithms and chain rule, we end up with \( \frac{1}{x} \).In essence, natural logarithms allow us to transform multiplicative processes into additive ones, which simplifies many calculus operations and makes understanding complex mathematical phenomena easier.