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Calculating derivatives is a crucial skill in calculus. It involves finding the slope of the tangent line to a curve at any point. In simpler terms, it's about determining how a function changes at various points. A derivative can be thought of as a rate of change. For the function given as \( f(x) = \frac{\ln x}{x} \), we are tasked with finding its second derivative, which means differentiating it twice.

When you compute the first derivative, you often need to apply specific rules, like the power, product, or quotient rule, depending on the structure of the function. Once the first derivative is found, you then differentiate that result to obtain the second derivative.

• First Derivative: Provides the slope or velocity of the function.
• Second Derivative: Offers information about the concavity or acceleration of the function.

Understanding these concepts enables the analysis of functions for critical points, inflection points, and can even help in sketching their graphs.

The quotient rule is the tool used to differentiate functions that are divided by each other, such as \( \frac{\ln x}{x} \) in our problem. The basic formula is:
\[ \left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2} \]
Here, the function in the numerator is denoted by \( u \), and the function in the denominator is \( v \). Differentiating a quotient involves substituting the derivatives of these two functions into this formula. For our function, \( u = \ln x \) and \( v = x \). It follows that:

• \( u' = \frac{1}{x} \), the derivative of \( \ln x \).
• \( v' = 1 \), the derivative of \( x \).

Plugging these into the formula gives the first derivative \( f'(x) = \frac{1 - \ln x}{x^2} \).

This rule is particularly beneficial when dealing with functions that have a simple ratio, allowing you to efficiently determine how these quotients change.

Logarithmic differentiation offers a powerful technique when dealing with complex functions involving logarithms. It's useful for functions not easily differentiated using standard rules. While this particular exercise takes a direct approach using the quotient rule, understanding logarithmic differentiation expands your toolkit for future challenges.

When would you use logarithmic differentiation? Imagine dealing with products of functions or very complex quotients. You would:

• Take the natural logarithm on both sides of an equation, which simplifies multiplicative complexities by turning products into sums.
• Differentiating each side, leveraging easier rules like the derivative of \( \ln x \), which is \( \frac{1}{x} \).
• Substituting back any variables if needed to express the derivative in terms of the original function.

This methodological approach allows the handling of log-based functions with greater ease and can sometimes transform intimidating differentiations into approachable ones.