Problem 76 - Calculus and its applications

Question

Differentiate. $$ y=\ln \frac{x^{4}}{2} $$

Step-by-Step Solution

Answer

The derivative is \( \frac{4}{x} \).

Step 1: Identify the Function

The function we need to differentiate is given as \( y = \ln \frac{x^{4}}{2} \). This is a logarithmic function of the form \( y = \ln(u) \), where \( u = \frac{x^{4}}{2} \).

Step 2: Apply the Logarithm Property

We know that \( \ln \left(\frac{a}{b}\right) = \ln a - \ln b \). Apply this property to rewrite the function as \( y = \ln(x^{4}) - \ln(2) \). Given \( \ln(2) \) is a constant, its derivative will be zero.

Step 3: Differentiate \( \\ln(x^{4}) \)

To differentiate \( \ln(x^{4}) \), use the chain rule. Start with \( \frac{d}{dx}\ln(x^{n}) = \frac{n}{x} \), so \( \frac{d}{dx}\ln(x^{4}) = \frac{4}{x} \).

Step 4: Combine the Derivatives

Combine the derivatives obtained: \( \frac{d}{dx} [ \ln(x^{4}) - \ln(2) ] = \frac{4}{x} - 0 \). Since the derivative of a constant \( \ln(2) \) is zero, the result is \( \frac{4}{x} \).

Key Concepts

Logarithmic Differentiation

Logarithmic differentiation is a technique used for finding the derivatives of complex functions, especially those that are products or quotients of other functions. The idea is to take advantage of the properties of logarithms, which can simplify the differentiation process.

In the equation you're differentiating, \( y = \ln\left(\frac{x^{4}}{2}\right) \), logarithmic differentiation allows us to simplify the expression to something more manageable. By writing the logarithm in terms of easier-to-handle sums and differences, differentiation becomes much simpler.

For example, the quotient \( \frac{x^{4}}{2} \) changes to \( x^{4} - 2 \) using logarithmic properties, which we will discuss next, making it much simpler to apply standard differentiation rules. In general, logarithmic differentiation is particularly effective when the function involves challenging products or quotients that make standard differentiation cumbersome.

Chain Rule

The Chain Rule is a fundamental tool in calculus used to differentiate composite functions. It expresses the derivative of a composite function in terms of the derivatives of its inner and outer functions.

In the context of our function, \( y = \ln(x^{4}) \), the Chain Rule is crucial. Here, the function \( x^{4} \) is nested within the logarithmic function, making it a perfect candidate for the application of the Chain Rule.

When we differentiate \( \ln(x^{4}) \), we need to apply the Chain Rule: First, take the derivative of the outer function, which is the natural logarithm, resulting in \( \frac{1}{x^4} \). Then, multiply by the derivative of the inner function, \( x^{4} \), which is \( 4x^{3} \). Thus, the derivative of \( \ln(x^{4}) \) is \( \frac{4x^{3}}{x^{4}} = \frac{4}{x} \). This simplification is what makes the Chain Rule so powerful for differentiation.

Properties of Logarithms

The properties of logarithms are particularly useful in the differentiation process, especially when dealing with complex expressions that involve divisions or products. These properties help simplify such expressions, making differentiation straightforward.

Some of the key properties include:

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

In your exercise, we applied the Quotient Rule to separate the logarithm of a division into a difference: \( \ln\left(\frac{x^{4}}{2}\right) = \ln(x^{4}) - \ln(2) \).

This separation is vital because it isolates the variable part of the expression from any constant terms. While \( \ln(2) \) remains constant and its derivative becomes zero, the variable part \( \ln(x^{4}) \) becomes simple enough to differentiate using the Chain Rule. By harnessing these properties, the differentiation process is much cleaner and more approachable.