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The chain rule is a fundamental technique used in calculus to differentiate complex functions. When dealing with functions nested within other functions, like our example with \(\ln(1-3x)\), the chain rule becomes essential. In simple terms, the chain rule states: If you have a function \(g(x)\) inside another function \(f(u)\), where \(u = g(x)\), then the derivative is: \[\frac{d}{dx}[f(g(x))] = f'(g(x))g'(x).\]

This means you first differentiate the outer function, treating the inner function as a single entity, and then multiply it by the derivative of the inner function. Here, \(\ln(1-3x)\) is treated as the outer function, and \(1-3x\) is the inner one. We differentiated \(\ln(1-3x)\) as \(\frac{1}{1-3x}\), and then multiplied by the derivative of \(1-3x\), which is \(-3\). Thus, we got \(\frac{-3}{1-3x}\).

• The chain rule is particularly helpful when differentiating composed functions.
• It allows us to break down complex problems into simpler steps.
• Practice using the chain rule to become more proficient in calculus.

The change of base formula is an essential tool when working with logarithmic functions, especially if the base of the logarithm is not natural, as in the case of \(\log_{2}(1-3x)\). By converting it to an expression involving natural logarithms, differentiation becomes more straightforward.

The formula is given as: \[\log_{b}(a) = \frac{\ln(a)}{\ln(b)}.\]
This allows us to express the base-2 logarithm in terms of \(\ln(1-3x)\) over \(\ln 2\). By applying this rearrangement, differentiation involves natural logs, which benefit from well-understood properties. After conversion, only the numerator function (the \(\ln(1-3x)\) part) needs differentiation, greatly easing the process.

• Make logarithm bases consistent to simplify differentiation.
• Use properties of logs to help manage complex derivatives.
• Efficiency improves with practice in using this formula.

Natural logarithms, denoted as \(\ln\), use the number \(e \approx 2.71828\) as their base, and they're prevalent in calculus due to their unique properties. When differentiating, the natural log has a simple derivative rule, \(\frac{d}{dx} \ln(u) = \frac{1}{u} \cdot \frac{du}{dx}\). This property makes working with \(\ln\) straightforward. In our exercise, the function \(\ln(1-3x)\) is central after applying the change of base.

With \(\ln\), it's important to analyze its behavior with transformations inside the argument, like \(1-3x\). In this context, the natural log's derivative involves applying the chain rule, where the external \(\ln\) function influences the result. Once we establish \(\ln(1-3x)\), we can easily proceed through the necessary calculus steps.

• Natural logs are key tools in solving calculus problems.
• They offer convenient properties for differentiation.
• Understanding \(\ln\) aids in tackling more complex mathematical tasks.