Problem 13 - Calculus Single Variable

Question

In each of Exercises \(13-30,\) calculate the derivative with respect to \(x\) of the given expression. \(\ln (4 x)\)

Step-by-Step Solution

Answer

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

Step 1: Identify the Function

The given function is \(y = \ln(4x)\). This is a composite function where the natural logarithm function \(\ln(u)\) is applied to \(u = 4x\).

Step 2: Use the Chain Rule

The derivative of a composite function, where \(y = \ln(u)\), is given by \(\frac{dy}{dx} = \frac{1}{u} \cdot \frac{du}{dx}\). First, find the derivative of the outer function \(\ln(u)\) with respect to \(u\); it is \(\frac{1}{u}\).

Step 3: Differentiate the Inner Function

The inner function is \(u = 4x\). Differentiate it with respect to \(x\): \(\frac{du}{dx} = 4\).

Step 4: Combine Using the Chain Rule

Substitute the derivatives from the previous steps into the chain rule formula: \[ \frac{dy}{dx} = \frac{1}{4x} \cdot 4 = \frac{4}{4x} \].

Step 5: Simplify the Expression

Cancel common factors in the derivative expression: \(\frac{4}{4x} = \frac{1}{x}\). This is the derivative of the given function.

Key Concepts

Composite Functions

In mathematics, composite functions consist of two or more functions where the output of one function becomes the input of another. Understanding composite functions is crucial for learning how to apply the chain rule in calculus.
Consider the given function: \(y = \ln(4x)\). Notice here, \(\ln(u)\) is our outer function and represents the natural logarithm, while \(u = 4x\) is the inner function. This inner function acts as the argument for the logarithmic function. Composite functions provide a layered approach to problem-solving.

• Outer function: \(\ln(u)\)
• Inner function: \(u = 4x\)

The joy of working with composite functions is that they grant us the flexibility to tackle complex expressions by breaking them down into simpler parts. This decomposition allows mathematicians and students alike to see the mechanisms of the function and to apply differentiation techniques like the chain rule eloquently.

Natural Logarithm

The natural logarithm, often written as \(\ln(x)\), usually refers to the logarithm to the base \(e\) (where \(e\) is approximately 2.71828). It is a fundamental mathematical function used extensively in calculus and scientific computations.
When differentiating functions involving natural logarithms, it is crucial to recognize the natural logarithm has a straightforward derivative formula:

• Derivative of Natural Logarithm: \[\frac{d}{dx} (\ln(x)) = \frac{1}{x}\]

This formula tells us that the slope of the tangent line to the curve at any point \(x > 0\) is the reciprocal of \(x\). This property of the natural logarithm makes it particularly convenient for calculus operations, especially in dealing with exponential growth and decay scenarios. When integrated into composite functions, the natural logarithm part of the formula focuses primarily on applying the chain rule accurately.

Chain Rule

The chain rule is a critical tool in calculus used for finding the derivative of composite functions. When a function is composed of another function, the chain rule provides a method to differentiate both layers to find the overall derivative.
In simpler terms, if you have a function \(y\) that is a composition of two functions \(f(g(x))\), with an outer function \(f\) and an inner function \(g\), the derivative is found by:

• Taking the derivative of the outer function with respect to the inner function, denoted as \(f'(g(x))\).
• Multiplying the result by the derivative of the inner function, expressed as \(g'(x)\).

For the provided exercise with \(y = \ln(4x)\), using the chain rule involved:

• Recognizing the derivative of \(\ln(u)\), which is \(\frac{1}{u}\),
• And the derivative of \(u = 4x\), which is \(4\).

By multiplying these derivatives together as per the chain rule, the exercise showed that the overall derivative simplifies nicely to \(\frac{1}{x}\). Embracing the chain rule empowers mathematicians to differentiate complex nested functions effectively.