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When we talk about the derivative of a logarithmic function, it's essential to understand the basic formula first. If you have a function of the form \( f(x) = \log_b(u) \), where \( u \) is some function of \( x \), the derivative takes a specific form. Instead of working directly with the function, we'll focus on its rate of change.
This rate of change is known as the derivative. For a logarithmic function, the derivative \( f'(x) \) is expressed as:

• \( f'(x) = \frac{1}{u \ln(b)} \cdot u' \)

Here, \( u \) is the function inside the logarithm, and \( u' \) is its derivative. The natural logarithm of \( b \), \( \ln(b) \), is a part of adjusting the base of the logarithm for differentiation. This formula helps us find the derivative of any logarithmic function if we know the inner function and its derivative.

The chain rule is a core principle in calculus that lets us find the derivative of composite functions. It's like peeling layers of an onion; you deal with the outer layer before the inner one. If a function is composed of two or more functions intertwined, the chain rule guides us through the differentiation process.
Take a composite function \( f(x) = \log_b(3x^2 - 2) \). The outer function here is the logarithm \( \log_b \) and the inner function is \( 3x^2 - 2 \). To differentiate, we:

• First, differentiate the outer function while keeping the inner function unchanged.
• Then, multiply by the derivative of the inner function (in our case, the derivative of \( 3x^2 - 2 \), which is \( 6x \)).

Using the chain rule helps us correctly structure the derivative expression, ensuring we account for each function involved.

Solving calculus problems is often a structured process. This means following clear steps to reach a solution. Let's look at a framework we can use:

• **Understand the problem:** Start by carefully reading what's being asked. Here, we need \( f'(1) = 3 \) for a specific function.
• **Break down the function:** Identify individual function parts and expressions, such as inner and outer functions.
• **Apply relevant calculus rules:** Use rules like the chain rule to differentiate effectively.
• **Substitute values:** Plug in numbers, such as \( x = 1 \), to simplify expressions where needed.
• **Solve for unknowns:** Here, it means manipulating the equation to solve for \( b \).
• **Verify your solution:** Always double-check calculations to ensure the answer makes logical sense.

Following these problem-solving steps ensures we systematically address each part, reducing errors and enhancing understanding.