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Differentiation is a fundamental concept in calculus. It involves finding the rate at which a function is changing at any given point. Simply put, it is the process of finding the derivative of a function.
When differentiating, we pay attention to how each part of the function contributes to the overall change. The result of differentiation is a new function, called the derivative, which gives the slope of the original function at any point.
In practice, differentiation applies various rules to determine the derivative of complicated functions by breaking them down into simpler parts.

Logarithmic functions are the inverses of exponential functions. The most common logarithmic function in calculus is the natural logarithm, denoted as \ln and having the base \e (Euler's number, approximately equal to 2.71828).
Logarithmic functions have special properties that make them useful in differentiation. Notably, the natural logarithm of a product can be separated into a sum: \( \ln(ab) = \ln(a) + \ln(b) \). This property makes it easier to differentiate complex logarithmic functions by breaking them down into simpler terms.
For example, in the given exercise, the function \( y = 3 \ln x + \ln 2 \) consists of natural logarithmic terms which can be individually differentiated and then combined.

To differentiate functions effectively, we use some commonly known rules:

• Power Rule: If \( y = x^n \), then \( \frac{dy}{dx} = nx^{n-1} \).
• Product Rule: If \( y = uv \) where both \(u\) and \(v\) are functions of \(x\), then \( \frac{dy}{dx} = u'v + uv' \).
• Quotient Rule: If \( y = \frac{u}{v} \), then \( \frac{dy}{dx} = \frac{u'v - uv'}{v^2} \).
• Chain Rule: If \( y = f(g(x)) \), then \( \frac{dy}{dx} = f'(g(x))g'(x) \).

For logarithmic functions, the key rule is: the derivative of \( \ln x \) with respect to \( x \) is \( \frac{1}{x} \).
In our exercise, using this rule for the term \(3 \ln x\), we get \( 3 \cdot \frac{1}{x} \). The other term, \( \ln 2 \), is a constant, so its derivative is zero. Therefore, the combined result is \( \frac{3}{x} \).