Question

$$ \text { } , \text { find the indicated derivative. } $$ $$ D_{x} \ln \sqrt{3 x-2} $$

Short answer

The derivative is \( \frac{3}{2(3x - 2)} \).

Step-by-step solution

Understand the Problem

We need to find the derivative of the function \( \ln \sqrt{3x - 2} \) with respect to \( x \). This involves using the chain rule and properties of logarithms and exponents.

Rewrite the Expression

Rewrite the function \( \ln \sqrt{3x - 2} \) using the properties of logarithms and exponents. Recall that \( \sqrt{u} = u^{1/2} \), therefore, \( \ln \sqrt{3x - 2} = \ln ((3x - 2)^{1/2}) \). Using the logarithmic identity \( \ln(a^b) = b \ln(a) \), we have \( \frac{1}{2} \ln(3x - 2) \).

Apply the Chain Rule

To find \( D_x \left( \frac{1}{2} \ln(3x - 2) \right) \), apply the chain rule. The derivative of \( \ln(u) \) with respect to \( u \) is \( \frac{1}{u} \). Therefore, the derivative of \( \frac{1}{2} \ln(3x - 2) \) with respect to \( x \) is \( \frac{1}{2} \cdot \frac{1}{3x - 2} \cdot (3) \) because the derivative of the inside function \( 3x - 2 \) is \( 3 \).

Simplify the Derivative

Simplify the expression: \( \frac{1}{2} \cdot \frac{1}{3x - 2} \cdot (3) = \frac{3}{2(3x - 2)} \). This is the simplified form of the derivative of the given function with respect to \( x \).

Key concepts

Chain Rule

The chain rule is a fundamental concept in calculus, especially when dealing with composite functions. It allows us to differentiate a function that is the composition of two or more functions.
To apply the chain rule, consider two functions: an outer function and an inner function. For instance, if we have a function defined as:

• The outer function is \( f(u) = \ln(u) \)
• The inner function is \( u = 3x - 2 \)

The chain rule states that the derivative of the composition \( f(g(x)) \) is \( f'(g(x)) \, g'(x) \).
In our example, when finding the derivative of \( \ln(3x - 2) \), we take the derivative of the outer function (\( \ln(u) \)), which is \( \frac{1}{u} \), and multiply it by the derivative of the inner function (\( 3x - 2 \)), which is \( 3 \). Ultimately, it gives us \( \frac{3}{3x - 2} \).

Logarithmic Differentiation

Logarithmic differentiation is a powerful technique for differentiating functions that are complex or cumbersome to handle using standard differentiation rules. It proves particularly helpful when dealing with products, quotients, or powers of functions.
The technique often starts with taking the natural logarithm of both sides of an equation \( y = f(x) \). We then apply differentiation laws, taking advantage of the properties of logarithms to simplify our expression before differentiating.
In the exercise, \( \ln \sqrt{3x - 2} \) is rewritten using logarithmic differentiation by evaluating \( \ln((3x - 2)^{1/2}) \), which simplifies to \( \frac{1}{2} \ln(3x - 2) \). This rewrite makes the differentiation process smoother by reducing the expression to a simpler form.

Properties of Logarithms

Understanding the properties of logarithms is essential for manipulating logarithmic expressions effectively. These properties help simplify complicated expressions into manageable forms. Some key properties include:

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

These properties are fundamental in the exercise when we transform \( \ln \sqrt{3x - 2} \) to \( \frac{1}{2} \ln(3x - 2) \) using the power property. Breaking down expressions with these properties makes the differentiation process straightforward and more intuitive.

Exponential Functions

Exponential functions are pervasive in calculus and have peculiar characteristics that influence their differentiation. These functions generally take the form of \( a^x \) or \( e^x \), where \( e \) is Euler’s number. Differentiation of exponential functions follows specific rules.
For instance, when differentiating \( e^{f(x)} \), we apply the chain rule where the derivative is \( e^{f(x)} \times f'(x) \). When applying these principles, it is crucial to combine them with logarithmic identities to handle complexities, such as when converting a radical expression to an exponential expression before differentiation.
In our context, expressions like \( \sqrt{} \) can be rewritten as exponential forms, making exponential rules applicable. Doing so aids in simplifying the differentiation process, as seen when \( \sqrt{3x - 2} \) is rewritten to \( (3x - 2)^{1/2} \), streamlining the derivative's calculation.