Problem 21 - Calculus and its applications

Question

Differentiate. $$ y=\log _{9}\left(x^{4}-x\right) $$

Step-by-Step Solution

Answer

\( \frac{d}{dx} = \frac{4x^3 - 1}{(x^4 - x) \ln(9)} \)

Step 1: Recall the Logarithmic Differentiation Formula

To differentiate a logarithm with a base other than e, recall that if a function is of the form \( y = \log_b(u) \), its derivative is \( \frac{d}{dx}[\log_b(u)] = \frac{1}{u \ln(b)} \cdot \frac{du}{dx} \). For this exercise, \( b = 9 \) and \( u = x^4 - x \).

Step 2: Differentiate the Argument of the Logarithm

Find the derivative of the function inside the logarithm. The argument \( u = x^4 - x \) differentiates to \( \frac{du}{dx} = 4x^3 - 1 \).

Step 3: Apply the Logarithmic Differentiation Formula

Substitute \( u \) and \( \frac{du}{dx} \) into the derivative formula from Step 1. This gives \( \frac{d}{dx}\left[\log_9(x^4 - x)\right] = \frac{1}{(x^4 - x) \ln(9)} \cdot (4x^3 - 1) \).

Step 4: Simplify the Expression

Simplify the expression to get the final derivative: \( \frac{d}{dx} = \frac{4x^3 - 1}{(x^4 - x) \ln(9)} \). This is the derivative of \( y = \log_9(x^4 - x) \).

Key Concepts

Derivative of Logarithms

Understanding the derivative of logarithms is essential when working with logarithmic functions in calculus. To derive a logarithmic function, especially one with a non-natural base, we use a specific formula that helps us calculate the derivative effectively.
For a logarithm with a base of \(b\), such as \(y = \log_b(u)\), the derivative is determined with the formula:

• \( \frac{d}{dx}[\log_b(u)] = \frac{1}{u \ln(b)} \cdot \frac{du}{dx} \)

This equation illustrates how to handle two parts:

• The function inside of the logarithm, \( u \), and
• The base of the logarithm, \(b\), expressed using a natural logarithm (\( \ln(b) \)).

To differentiate such logarithmic expressions means you need to know both these components well, and then proceed to calculate using the given formula.

Chain Rule

The chain rule is a powerful technique used in differentiation, important for dealing with composite functions. When you have one function embedded within another, the chain rule lets you find the derivative of the overall function.
In our logarithmic differentiation example, the chain rule comes into play when we differentiate the inner function \( u = x^4 - x \).Here's how the chain rule manifests:

• Differentiate the outer function \( \log_9(u) \) with respect to \( u \), which involves \( \frac{1}{u \ln(9)} \).
• Next, differentiate the inner function \( u = x^4 - x \) to get \( 4x^3 - 1 \).

You then multiply these derivatives to find the derivative of the entire composite function. This step-by-step differentiation ensures precision and clarity when handling more complex functions.

Differentiation Techniques

Differentiating functions requires knowledge of various techniques, each suited to different types of expressions. For logarithmic functions and their derivatives, mastering techniques like the chain rule and understanding logarithm principles are crucial.
Consider these steps and strategies:

• Logarithmic Differentiation: Use this specifically for expressions involving logs, especially with non-base \(e\) logs.
• The Power Rule Use: When dealing with polynomial expressions like \( x^4 - x \), differentiate each term separately to find \( 4x^3 - 1 \).
• Simplification: After differentiating, always simplify the expression to find a clean and comprehensive derivative result.

These techniques collectively help in obtaining the derivative of a function like \( y = \log_9(x^4 - x) \), offering a streamlined approach to solve such calculus problems effectively.