Problem 19 - University Calculus: Early Transcendentals

Question

Find the derivative of \(y\) with respect to \(x, t,\) or \(\theta\) as appropriate. $$y=\ln x^{3}$$

Step-by-Step Solution

Answer

Derivative is \(\frac{3}{x}\).

Step 1: Identify the function and rewrite if necessary

The given function is \(y = \ln x^3\). We'll start by rewriting this using the logarithmic property \(\ln a^b = b \ln a\). So, we have \(y = 3 \ln x\).

Step 2: Differentiate using basic rules

Differentiate \(y = 3 \ln x\) with respect to \(x\). Recall the derivative of \(\ln x\) is \(\frac{1}{x}\). Using this, we get \(\frac{dy}{dx} = 3 \cdot \frac{1}{x} = \frac{3}{x}\).

Key Concepts

Logarithmic Differentiation

Logarithmic differentiation is a technique that helps simplify complex differentiation problems. It is particularly useful when dealing with functions where the variable has an exponent. The main idea is to take the natural logarithm of both sides of the equation, apply properties of logarithms to simplify, and then differentiate.
Here's why it's so powerful: sometimes a function is too complicated to differentiate directly. Logarithmic differentiation turns multiplication and division into addition and subtraction, which are much easier to deal with. This approach can even handle products of many functions or power functions neatly.
For example, in the function \(y = \ln x^3\), the exponent is brought down using the power rule of logarithms, leading to a more straightforward expression for differentiation. This simplification prepares the ground for applying basic differentiation rules effectively.

Properties of Logarithms

The properties of logarithms are essential mathematical rules that help simplify expressions involving logarithms. They allow you to break down complex functions into more manageable parts. Here are a few key properties that are frequently used:

• Power Rule: \(\ln a^b = b \ln a\). This lets you bring down the exponent, simplifying the process of differentiation. For instance, \(\ln x^3 = 3 \ln x\).
• Product Rule: \(\ln(ab) = \ln a + \ln b\). This breaks down the logarithm of a product into a sum, making it easier to handle.
• Quotient Rule: \(\ln\left( \frac{a}{b} \right) = \ln a - \ln b\). This rule is used to simplify the logarithm of a quotient.

Using these properties allows for the simplification of the differentiation process, as seen in the example function. By rewriting \(y = \ln x^3\) as \(y = 3 \ln x\), differentiation is simplified.

Basic Differentiation Rules

When it comes to calculus, differentiation is one of the foundational principles. Understanding the basic differentiation rules is crucial for finding the derivative of functions effectively.
In our example, where \(y = 3 \ln x\), we use the rule that the derivative of \(\ln x\) is \(\frac{1}{x}\). Basic differentiation rules also include:

• Power Rule: If \(y = x^n\), then \(\frac{dy}{dx} = nx^{n-1}\).
• Constant Rule: The derivative of a constant is zero.
• Constant Multiple Rule: If a function is multiplied by a constant, the constant can be factored out of the differentiation, such as in \(\frac{d}{dx}[cu]=c\frac{du}{dx}\).
• Sum Rule: The derivative of a sum is the sum of the derivatives.

Applying these rules lets you easily find \(\frac{dy}{dx} = 3 \cdot \frac{1}{x}\), simplifying to \(\frac{3}{x}\). Thus, mastering these basic rules is essential for solving more complex calculus problems.