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When it comes to differentiating composite functions – functions made up of two or more functions – the chain rule is an indispensable tool. It allows us to differentiate a function of a function by taking the derivative of the outside function and multiplying it by the derivative of the inside function.

For example, consider the function \(h(x) = (g(f(x)))\). The chain rule tells us that \(h'(x) = g'(f(x)) \cdot f'(x)\). It's like peeling an onion – you differentiate layer by layer. This becomes especially useful when dealing with exponential functions, where the exponent itself is a function of \(x\).

In our exercise, \(y = 5 \cdot 4^x\), the power \(x\) is the inner function, and the exponential part \(4^x\) is the outer function. Applying the chain rule isn't necessary in this particular example as the inner function is simply \(x\), which has a derivative of 1. However, understanding the chain rule is crucial for tackling more complex exponential functions where the exponent is not just \(x\), but a function of \(x\) itself.

The exponential function rule is a specific case of the chain rule applied to exponential functions. This rule states that the derivative of an exponential function with a base \(a\) is the original function multiplied by the natural logarithm of the base, \(a\). Mathematically, this is expressed as \(\frac{d}{dx}(a^x) = a^x \cdot \ln(a)\).

Why the natural logarithm, though?

It relates to the fact that the rate of change of the exponential function \(a^x\) is directly proportional to the function itself – and the constant of proportionality is \(\ln(a)\). In our exercise, we applied this rule to differentiate \(4^x\), ending up with \(\frac{d}{dx}(4^x) = 4^x \cdot \ln(4)\).

When a function is multiplied by a constant, the constant multiple rule makes differentiation straightforward. Simply put, the derivative of a constant times a function is the constant times the derivative of the function. The rule is formally written as \(\frac{d}{dx}(c \cdot f(x)) = c \cdot \frac{d}{dx}(f(x))\).

This means if we have \(y = c \cdot f(x)\) and \(c\) is a constant, we can pull \(c\) out in front of the derivative and focus on differentiating \(f(x)\) alone. For our problem, \(y = 5 \cdot 4^x\), where \(5\) is a constant, we applied this rule to find \(\frac{d}{dx}(5 \cdot 4^x) = 5 \cdot \frac{d}{dx}(4^x)\).

The natural logarithm, denoted as \(\ln\), is the logarithm to the base \(e\), where \(e\) is approximately 2.71828. It's a special logarithm because of its relationship to exponential growth and decay problems, where the base of the exponential function is \(e\). The natural logarithm reverses the operation of raising \(e\) to a power, which means that \(\ln(e^x) = x\).

When we differentiate exponential functions with bases other than \(e\), the natural logarithm comes into play, as seen in the exponential function rule. For the given exercise, we used the natural logarithm of 4, \(\ln(4)\), to express the derivative of \(4^x\). The presence of the natural logarithm in derivatives of exponential functions underscores its importance in calculus, particularly in understanding growth and decay processes.