Problem 69 - Thomas Calculus in SI Units

Question

Find the derivative of \(y\) with respect to the given independent variable. $$y=\log _{4} x+\log _{4} x^{2}$$

Step-by-Step Solution

Answer

The derivative is \(\frac{3}{x \ln 4}\).

Step 1: Apply the properties of logarithms

The given expression is \(y = \log_{4} x + \log_{4} x^{2}\). According to the properties of logarithms, we can combine these logarithms into a single expression. The property \(\log_b a + \log_b c = \log_b (a \cdot c)\) allows us to write it as \(y = \log_{4}(x \cdot x^{2}) = \log_{4}(x^{3})\).

Step 2: Convert the logarithmic expression to natural logarithm

We can express \(\log_{4}(x^{3})\) in terms of natural logarithms because it simplifies the differentiation process. \(\log_{4}(x^{3}) = \frac{\ln(x^{3})}{\ln(4)} = \frac{3 \ln x}{\ln 4}\). Now, the expression becomes \(y = \frac{3 \ln x}{\ln 4}\).

Step 3: Differentiate with respect to \(x\)

Apply the chain rule to differentiate \( \frac{3 \ln x}{\ln 4} \) with respect to \(x\). The derivative of \(\ln x\) with respect to \(x\) is \(\frac{1}{x}\). Thus, the derivative \(\frac{dy}{dx} = \frac{3}{\ln 4} \cdot \frac{1}{x} = \frac{3}{x \ln 4}\).

Key Concepts

Logarithmic Differentiation

Logarithmic differentiation is a technique used to differentiate functions that are difficult to handle using standard differentiation rules. It involves taking the logarithm of both sides of an equation and then differentiating. This method is particularly useful for complex products, quotients, or powers, where the log transformation simplifies the differentiation process.

When you apply a logarithm to a product, quotient, or power, the logarithmic properties allow you to turn these into sums, differences, and products, which are typically much easier to differentiate. In our problem, the expression is initially of the form \( y = \log_{4} x + \log_{4} x^{2} \). By using the properties of logarithms, it becomes \( y = \log_{4} (x^3) \), simplifying the overall expression.

This approach reduces complex chains of products or powers into single variables or simpler forms, making the subsequent steps in differentiation much more straightforward.

Chain Rule

The chain rule is a fundamental tool in calculus for differentiating compositions of functions. If you have a function nested inside another, the chain rule helps you find the derivative of the outer function with respect to the inner function.

In the expression \( y = \frac{3 \ln x}{\ln 4} \), where \( y \) is expressed in terms of \( \ln x \), the chain rule helps us differentiate \( \ln x \). We understand that the derivative of \( \ln x \) is \( \frac{1}{x} \). Combined with the constant multiplier rule, the derivative of \( 3 \ln x \) is simply \( \frac{3}{x} \).

This application of the chain rule is a classic example where we differentiate each component of a function individually and then multiply by the derivative of the respective inner functions. This simplifies otherwise complicated expressions.

Logarithmic Properties

Logarithmic properties are essential tools in simplifying expressions and performing differentiations. They allow you to manipulate logs into simpler forms, often transforming products into sums and powers into products.

In our problem, we start with \( y = \log_{4} x + \log_{4} x^{2} \). Using the property \( \log_b a + \log_b c = \log_b (a \cdot c) \), we can combine the two log terms: \( \log_{4}(x \cdot x^2) = \log_{4}(x^{3}) \). This step condenses the problem into a more manageable form.

Moreover, changing the base of logarithms is another useful property. We use \( \log_{b}(a) = \frac{\ln(a)}{\ln(b)} \) to express the log in terms of natural logarithms, which are easier to differentiate: \( \log_{4}(x^{3}) = \frac{3 \ln x}{\ln 4} \). By applying these properties, differentiation becomes a simpler task, highlighting just how powerful these logarithmic manipulations can be.