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Logarithms have specific properties that make them easier to manipulate and calculate. Understanding these properties is crucial when simplifying logarithmic expressions. Here are the main properties we used in simplifying the function \(f(x) = \ln \sqrt{10x}\).

• Product Property: This states that the logarithm of a product is the sum of the logarithms of each factor. In formula terms, \(\ln(ab) = \ln a + \ln b\).
• Power Property: This indicates that the logarithm of a power can be expressed as the exponent times the logarithm of the base. Expressed as \(\ln (a^r) = r \ln a\).

By applying these properties, we transformed \(\ln \sqrt{10 \cdot x}\) into \(\frac{1}{2}\ln 10 + \frac{1}{2}\ln x\). Each step relies on breaking down the components of the expression into more manageable parts, which can then be handled individually. This process is handy in differentiation, where simplifying complex terms can make finding derivatives much simpler.
This simplification allows a clear path to compute the derivative without being tangled in more complicated arithmetic.

Differentiation is a mathematical process used to calculate the rate at which a function is changing at any given point. It's a fundamental concept in calculus, allowing us to determine the slope of a function at any point, often referred to as the derivative of the function.
For our function, after simplifying it to \(f(x) = \frac{1}{2}\ln 10 + \frac{1}{2}\ln x\), we notice that the term \(\frac{1}{2}\ln 10\) is constant. Thus, its derivative is zero because a constant doesn’t change with respect to \(x\).
The primary focus then is on differentiating \(\frac{1}{2}\ln x\). The derivative of \(\ln x\) is \(\frac{1}{x}\). Multiplying by \(\frac{1}{2}\), as per the multiplication property, we find the derivative of this segment is \(\frac{1}{2x}\).
This calculation helps us understand how quickly the function \(f(x) = \ln \sqrt{10x}\) increases or decreases as \(x\) changes.

The chain rule is a critical differentiation method that facilitates finding the derivative of functions composed of other functions. Generally, it applies when differentiating a composite function by recognizing the inherent relationship between derivatives of the inner and outer functions.
In our exercise, even though it may not appear in its traditional form, the chain rule's logic is employed subconsciously. When computing the derivative of \(\frac{1}{2}\ln x\), understand that \(\ln x\) itself is a function.\ Every element in the derivative process builds upon understanding how functions interlink.
By systematically applying the chain rule, we divide the problem into simpler parts — like the way a ripe banana is easier to peel and eat in segments. The rule is indispensable when dealing with layered expressions, ensuring we account for all dimensions of change.