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The chain rule is a fundamental tool in calculus for finding the derivative of composite functions. It allows us to differentiate functions nested within others by breaking them down into simpler parts. In general, if you have a function composed as \( f(g(x)) \), the chain rule states that its derivative is \( f'(g(x)) \cdot g'(x) \). This means you first differentiate the outer function and then multiply it by the derivative of the inner function.

To apply the chain rule to the exercise given where \( y = (\ln x)^3 \), we first identify the inner and outer functions. Set \( u = \ln x \) (the inner function), which makes the function \( y = u^3 \) (the outer function). This setup simplifies differentiation:

• Differentiate the outer function: \( \frac{d}{du}(u^3) = 3u^2 \).
• Differentiate the inner function: \( \frac{du}{dx} = \frac{1}{x} \).
• Combine using the chain rule: \( \frac{dy}{dx} = 3(\ln x)^2 \cdot \frac{1}{x} \).

This results in the derivative \( \frac{3(\ln x)^2}{x} \).

Finding where a derivative equals zero can tell us about the critical points of the function, which are potential maximums, minimums, or points of inflection. To do this, set the derivative equal to zero and solve for the variable.

In the provided problem, the derivative \( \frac{dy}{dx} = \frac{3(\ln x)^2}{x} \) is set to zero to locate these points:

• The equation \( \frac{3(\ln x)^2}{x} = 0 \) simplifies to \( (\ln x)^2 = 0 \) because \( x eq 0 \).

By solving \( (\ln x)^2 = 0 \), we find that \( \ln x = 0 \) since a square root only equals zero if the original number was zero.

The natural logarithm \( \ln x = 0 \) occurs when \( x = e^0 = 1 \). Therefore, the derivative is zero at \( x = 1 \), indicating a critical point at this location.

The natural logarithm, denoted as \( \ln \), is the logarithm to the base \( e \), where \( e \) is an irrational and transcendental number approximately equal to 2.71828. It is widely used in calculus and mathematical analysis due to its unique properties.

When differentiating functions that involve \( \ln x \), you should remember that its derivative is simply \( \frac{1}{x} \). This makes working with logarithmic functions straightforward when applying rules such as the chain rule.

For example, in our exercise, \( y = (\ln x)^3 \) involves differentiating an expression raised to a power, which is streamlined by understanding the property of \( \ln x \) and its derivative:

• Express \( y \) in terms of \( u = \ln x \), making the differentiation process clearer.
• Apply the chain rule effectively knowing \( \frac{du}{dx} = \frac{1}{x} \).
• When solving \( \ln x = 0 \), conclude \( x = 1 \) since \( e^0 = 1 \).

With this foundation, solving complex problems involving natural logarithms becomes easier and more intuitive.