Problem 29 - Calculus Single Variable

Question

Calculate the derivative with respect to \(x\) of the given expression. \(\log _{2}(3 x)\)

Step-by-Step Solution

Answer

The derivative is \( \frac{1}{x \ln(2)} \).

Step 1: Understand the Logarithmic Derivative

The derivative of a logarithmic function with a base other than the natural base can be calculated using the change of base formula: \( \log_a(x) = \frac{\ln(x)}{\ln(a)} \). Thus, the logarithm \( \log_{2}(3x) \) can be rewritten as \( \frac{\ln(3x)}{\ln(2)} \).

Step 2: Apply the Quotient Rule

The derivative of \( \frac{\ln(3x)}{\ln(2)} \) can be found by treating \( \ln(2) \) as a constant multiplier. Therefore, differentiate \( \ln(3x) \) times \( \frac{1}{\ln(2)} \).

Step 3: Differentiate \( \ln(3x) \)

Use the chain rule to differentiate \( \ln(3x) \). We have: \( \frac{d}{dx}[\ln(3x)] = \frac{1}{3x} \times 3 = \frac{1}{x} \).

Step 4: Calculate the Derivative

Combine the derivative from Step 3 with the constant from Step 2: \( \frac{d}{dx}[\log_2(3x)] = \frac{1}{\ln(2)} \times \frac{1}{x} = \frac{1}{x \ln(2)} \).

Key Concepts

Change of Base Formula

When dealing with logarithms, sometimes you encounter a base that is not convenient for differentiation or computation. Enter the **Change of Base Formula**, which is a way to convert logarithms from any base to another, typically the natural logarithm base, which is much easier to handle in calculus.

The Change of Base Formula states:

• \( \log_a(x) = \frac{\ln(x)}{\ln(a)} \)

This formula is quite handy because it allows you to transform the logarithm into a fraction involving natural logarithms. In our exercise, this was used to rewrite \( \log_{2}(3x) \) as \( \frac{\ln(3x)}{\ln(2)} \). Here, \( \ln(3x) \) becomes the primary expression to differentiate using other calculus rules.

This transformation is the first critical step, allowing for the application of more straightforward differentiation techniques available for natural logs.

Quotient Rule

Once we rewrite a logarithm using the Change of Base Formula, differentiation might involve the **Quotient Rule**, specifically when the expression takes the form of a fraction.

The Quotient Rule is described as follows:

• If you have a function \( \frac{u}{v} \), then its derivative \( \left(\frac{u}{v}\right)' \) is \( \frac{u'v - uv'}{v^2} \).

However, in this particular problem, because \( \ln(2) \) is a constant, you do not need to use the full Quotient Rule. Instead, treat \( \ln(2) \) as a constant coefficient. Thus, the problem simplifies to multiplying the differentiation of \( \ln(3x) \) by \( \frac{1}{\ln(2)} \).

This method reduces the complexity of employing the Quotient Rule directly and helps you focus on the crucial part, which is differentiating the function \( \ln(3x) \).

Chain Rule

For the differentiation of composite functions, the **Chain Rule** is an essential tool. It is employed when you have a function within another function. For our exercise, this is evident in \( \ln(3x) \).

The Chain Rule states the following:

• If you have a composite function \( \ln(f(x)) \), then its derivative is \( f'(x) \times \frac{1}{f(x)} \).

For our function \( \ln(3x) \):

• The inside function is \(3x\).
• The derivative of \(3x\) is \(3\).

Accordingly, applying the Chain Rule gives:

• \( \frac{d}{dx}[\ln(3x)] = \frac{1}{3x} \times 3 = \frac{1}{x} \)

Ultimately, combining this result with the constant multiplier from the earlier steps gives the complete derivative for \( \log_{2}(3x) \). The Chain Rule ensures every component of the composite function is accurately differentiated, leading to an effective solution.