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The natural logarithm, denoted as \( \ln \), is the logarithm to the base \( e \), where \( e \) is an irrational number approximately equal to 2.71828. This special logarithm is commonly used in mathematics due to its unique properties which simplify many calculus operations. One significant property of the natural logarithm is its derivative. If you have a function \( \ln(u) \), where \( u \) is a function of \( t \), its derivative is \( \frac{1}{u} \times \frac{du}{dt} \). This formula allows us to find the rate at which the natural logarithm function changes with respect to time or any variable \( t \). Natural logarithms appear frequently in scientific calculations and help model exponential growth processes like population growth and radioactive decay. Understanding the derivative of \( \ln \) forms the basis for solving complex calculus problems involving logarithmic differentiation.

Differentiation is the process of finding the derivative of a function. In simple terms, a derivative represents the rate at which a function changes at any given point. It's equivalent to finding the slope of the tangent line to the function at a particular point. For example, consider the function \( y = f(t) \). The derivative \( f'(t) \) of \( y \) is determined by how \( y \) changes as \( t \) slightly increases. Differentiation can apply to constants, algebraic expressions, exponential functions, and more. In the given problem, we start with a simple differentiation of the constant term \( 5 \). The derivative of any constant is always zero because constants do not change and thus have no rate of change. Understanding how to differentiate each part of a composite function, like \( 5 + \ln(3t + 2) \), allows us to combine these parts successfully for a complete derivative.

The Chain Rule is a fundamental theorem in calculus that helps differentiate composite functions. Composite functions are combinations of two or more functions, where one function is nested within another, such as \( \ln(3t + 2) \). The Chain Rule states: if you have two functions \( f(g(x)) \), then the derivative \( f'(g(x)) \) is \( f'(g(x)) \cdot g'(x) \). Essentially, you differentiate the outer function and multiply by the derivative of the inner function. When applying the Chain Rule in the example problem, we treat \( u = 3t + 2 \) as the inner function and \( \ln(u) \) as the outer function. First, we differentiate the inner function \( u = 3t + 2 \) to get \( \frac{du}{dt} = 3 \). Then, we apply it to the derivative formula for \( \ln(u) \), resulting in the outer derivative being \( \frac{1}{3t + 2} \) multiplied by the inner \( \frac{du}{dt} \) which is 3, giving \( \frac{3}{3t + 2} \). The utilization of the Chain Rule simplifies working with nested functions and is crucial for accurate differentiation of complex expressions.