Problem 15 - Calculus Single Variable

Question

Calculate the derivative with respect to \(x\) of the given expression. \(\ln (2 x+3)\)

Step-by-Step Solution

Answer

The derivative of \( \ln(2x + 3) \) is \( \frac{2}{2x + 3} \).

Step 1: Identify the function

The function provided is \( f(x) = \ln(2x + 3) \). We want to calculate the derivative of this function with respect to \(x\).

Step 2: Recall the derivation rule for a natural logarithm

The derivative of the natural logarithm function \( \ln(u) \) with respect to \( x \) is given by \( \frac{1}{u} \frac{du}{dx} \).

Step 3: Identify inner function \(u\)

For the function \( f(x) = \ln(2x + 3) \), the inner function \( u \) is \( 2x + 3 \).

Step 4: Compute the derivative of the inner function \(u\)

Calculate \( \frac{du}{dx} \) where \( u = 2x + 3 \). The derivative of \( 2x + 3 \) with respect to \( x \) is \( 2 \).

Step 5: Combine results from previous steps

Using the rule \( \frac{d}{dx}[ \ln(u) ] = \frac{1}{u} \frac{du}{dx} \), substitute \( u = 2x + 3 \) and \( \frac{du}{dx} = 2 \) to get:\[ \frac{d}{dx}[ \ln(2x + 3)] = \frac{1}{2x + 3} \cdot 2 \].

Step 6: Simplify the expression

Simplify the expression \( \frac{2}{2x + 3} \) to obtain the final result.The derivative is \( \frac{2}{2x + 3} \).

Key Concepts

Natural Logarithm

The natural logarithm, often denoted as "ln," is a logarithmic function with a special base: the transcendental number \( e \), which is approximately equal to 2.71828. It is important because it naturally appears in various mathematical scenarios, particularly in calculus and exponential growth problems.
The natural logarithm has a unique property where the derivative of \( \ln(x) \) with respect to \( x \) is \( \frac{1}{x} \). This simplicity is a building block for many calculus operations, especially in differentiation and integration. When dealing with more complex functions such as \( \ln(2x+3) \), this property still applies, but it's necessary to take into consideration what is called the chain rule.
Using the natural logarithm is crucial when differentiating expressions involving products, quotients, and powers. It serves as a tool to "simplify" and "linearize" these expressions, making it easier to perform calculus operations such as differentiation and integration.

Chain Rule

The chain rule is a fundamental differentiation technique in calculus. It simplifies the process of finding the derivative of composite functions, which are functions made up of other functions. The rule essentially states that if you have a composite function \( f(g(x)) \), the derivative is \( f'(g(x)) \cdot g'(x) \).
In the specific context of the function \( f(x) = \ln(2x + 3) \), the outer function is \( \ln(u) \) and the inner function \( u = 2x + 3 \). The chain rule requires you to differentiate the outer function with respect to the inner function, then multiply by the derivative of the inner function.
Applying this rule:

• Differentiate \( \ln(u) \) to get \( \frac{1}{u} \).
• Find \( \frac{du}{dx} \) where \( u = 2x + 3 \), resulting in \( \frac{du}{dx} = 2 \).

Putting it all together, the derivative \( \frac{d}{dx}[ \ln(2x + 3)] = \frac{1}{2x + 3} \cdot 2 \). This technique showcases how composite functions are tackled by breaking them into manageable parts.

Function Differentiation

Function differentiation is a core concept in calculus that involves determining how a function changes with respect to its variable(s). Essentially, finding the derivative answers the question of how one quantity changes as another quantity changes.
When dealing with complex expressions such as \( \ln(2x + 3) \), understanding differentiation leads to insights about rates of change and is fundamental for solving problems in physics, economics, and beyond.
In differentiating mathematical functions:

• Identify the function type (exponential, logarithmic, polynomial, etc.).
• Apply relevant differentiation rules such as the power rule, product rule, quotient rule, or chain rule.
• Simplify the resulting expression if necessary, as seen in simplifying \( \frac{2}{2x + 3} \).

This systematic approach to function differentiation allows us to analyze curve behaviors, optimize solutions, and model real-world phenomena accurately.