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Differentiation of logarithmic functions often appears daunting at first glance, but it can be tamed with understanding. The function given in the exercise is a logarithmic expression: \( u = 2 \ln((3-x)^4) \). To differentiate this, we need to rely on some properties of logarithms.

One key property is the logarithmic power rule, which states that \( \ln(a^b) = b \ln(a) \). This allows us to bring the exponent down in front of the logarithm to simplify the differentiation process. Using this rule, the original function becomes:

• \( u = 2 \cdot 4 \ln(3-x) = 8 \ln(3-x) \)

Simplifying the function first makes the derivative easier to manage. This is an essential step when dealing with complex logarithmic expressions. The key takeaway is to recognize and utilize logarithmic properties for efficient differentiation.

The chain rule is a fundamental tool in calculus used to differentiate composite functions. If a function is the composition of two or more functions, the chain rule is employed. In this exercise, we reencounter this valuable tool.

When we have a function like \( 8 \ln(3-x) \), we first identify the outer and inner functions. Here, \( \ln(3-x) \) can be seen as the inner function, and multiplying by 8 is the outer operation. The derivative of a logarithmic function, \( \ln(x) \), is \( \frac{1}{x} \). But since the inner function is \( 3-x \), we apply the chain rule. The derivative of \( \ln(3-x) \) becomes \( \frac{-1}{3-x} \) because the derivative of \( 3-x \) is \(-1\).

• Outer function: \( 8 \)
• Inner function: \( \ln(3-x) \)

Combing these with the chain rule, we find the derivative: \(-8 \times \frac{1}{3-x} = \frac{-8}{3-x}\). This step highlights the chain rule's utility in breaking down and simplifying more challenging differentiation tasks.

Simplifying logarithms is a critical skill in calculus, particularly when preparing for differentiation or integration. By applying the properties of logarithms, we can tidy up expressions so they become significantly more manageable.

In the exercise, we utilized the logarithmic power rule to simplify \( \ln((3-x)^4) \) to \( 4 \ln(3-x) \). This simplification makes the differentiation process straightforward. Simplified functions not only reduce the potential for error but also enhance our understanding by stripping down to the essentials.

Always look out for opportunities to simplify expressions involving logarithms. This often involves:

• Applying the power rule: \( \ln(a^b) = b\ln(a) \)
• Using other properties like the product, quotient, or change of base rules when applicable

They can save time and simplify subsequent calculations. Keep practicing these techniques to make them second nature.