Problem 9 - Thomas Calculus

Question

In Exercises \(5-36,\) find the derivative of \(y\) with respect to \(x, t,\) or \(\theta,\) as appropriate. $$ y=\ln \frac{3}{x} $$

Step-by-Step Solution

Answer

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

Step 1: Rewrite the Original Function

The given function is \( y = \ln \left( \frac{3}{x} \right) \). To make differentiation easier, we can use the logarithm property that \( \ln \left( \frac{a}{b} \right) = \ln a - \ln b \). Thus, rewrite the function as \( y = \ln 3 - \ln x \).

Step 2: Differentiate the Function

Now, differentiate the expression \( y = \ln 3 - \ln x \) with respect to \( x \). Recall that the derivative of \( \ln x \) is \( \frac{1}{x} \). Since \( \ln 3 \) is a constant, its derivative is 0. Therefore, the derivative is \( \frac{dy}{dx} = 0 - \frac{1}{x} = -\frac{1}{x} \).

Key Concepts

Logarithmic Differentiation

Logarithmic differentiation is a helpful technique in calculus, especially useful when dealing with functions that are products, quotients, or powers. By introducing a logarithm, the complexity of differentiation can significantly reduce. This method uses the properties of logarithms to simplify differentiation, making calculations more manageable.
When faced with a quotient like \( y = \ln \left( \frac{3}{x} \right) \), logarithmic differentiation allows us to break it down into simpler parts using the property \( \ln \left( \frac{a}{b} \right) = \ln a - \ln b \). By rewriting the function, it is easier to take derivatives of simpler terms.
This method works particularly well on complex functions where direct differentiation would involve more complicated product or quotient rules, allowing us instead to use simpler rules that stem from properties of logarithms.

Differentiation Techniques

Differentiation techniques are essential tools in calculus that allow us to find the rate at which a function changes. One of the most commonly used techniques involves using properties of logarithms, as demonstrated in the logarithmic differentiation example.
To find the derivative of \( y = \ln \left( \frac{3}{x} \right) \), we can transform the function first and then differentiate. The derivative of \( \ln x \) is \( \frac{1}{x} \). Thus, the derivative of \( \ln \frac{3}{x} \) becomes the derivative of a difference: 0 for the constant \( \ln 3 \), and \( \frac{1}{x} \) for \( \ln x \). Therefore, \( \frac{dy}{dx} = -\frac{1}{x} \).
Using correct differentiation techniques ensures you can successfully handle various types of functions, especially those involving logarithms, exponentials, and trigonometric functions.

Properties of Logarithms

The properties of logarithms are powerful tools that can simplify the process of dealing with logarithmic functions in calculus. These properties convert complex expressions into simpler forms, making differentiation easier.
The key properties used in the differentiation example are:

• \( \ln(ab) = \ln a + \ln b \)
• \( \ln \left( \frac{a}{b} \right) = \ln a - \ln b \)
• \( \ln(a^b) = b \ln a \)

Using these properties, we can express a complex logarithmic function as a series of simpler terms. For instance, in \( y = \ln \left( \frac{3}{x} \right) \), utilizing the property \( \ln \left( \frac{a}{b} \right) = \ln a - \ln b \), the function becomes \( y = \ln 3 - \ln x \), making the differentiation process straightforward.
Mastering these properties can significantly ease the calculus process, especially when working with intricate expressions or equations.