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The chain rule is a fundamental concept in calculus used to differentiate composite functions. A composite function is a function of another function. For example, in our problem, we have the function

• \( e^{3x} \)
• \( \ln(3x) \)

Here, both contain inner functions:

• \( 3x \) inside \( e^{3x} \)
• \( 3x \) inside \( \ln(3x) \)

To apply the chain rule, first differentiate the outer function, then multiply by the derivative of the inner function.
For \( e^{3x} \), the derivative of \( e^u \) with respect to \( u \) is \( e^u \), while the derivative of the inner function \( 3x \) is \( 3 \). Therefore, the derivative is \( 3e^{3x} \).
Similarly, for \( \ln(3x) \), differentiate \( \ln(u) \) to obtain \( \frac{1}{u} \) and multiply by the derivative of \( 3x \) which is \( 3 \), obtaining \( \frac{3}{3x} = \frac{1}{x} \).
The chain rule simplifies the process of finding derivatives for complex expressions by using this step-by-step approach to tackle both the inner and outer components.

Exponential functions are ubiquitous in mathematics and sciences, providing a model for growth processes.
The general form for exponential functions is \( e^{f(x)} \), where \( e \) is the base of the natural logarithm, approximately equal to 2.718.
In calculus, these functions have a particularly straightforward derivative: the derivative of \( e^{u} \) is \( e^{u} \) itself, multiplied by the derivative of the exponent \( u \).

• For example, \( e^{3x} \) has an exponent \( 3x \). So like we calculated earlier, its derivative is \( 3e^{3x} \).

Exponential functions grow exceptionally fast, and understanding their rates of change, thanks to simple derivatives, is crucial in fields like finance, physics, and biology.
With the chain rule, handling the exponential growth of different functions becomes straightforward.

Logarithmic functions are the inverse of exponential functions and are vital in scenarios where growth slows down. They are represented by \( \ln(x) \) involving the natural logarithm.

• The derivative of a logarithmic function like \( \ln(u) \) is \( \frac{1}{u} \), which provides the slope of the function at any point.

However, note that we often encounter composite forms such as \( \ln(3x) \), which requires the chain rule to derive.
Calculating the derivative involves finding \( \frac{1}{3x} \) and multiplying by the inner derivative \( 3 \), resulting in \( \frac{1}{x} \).
The typical way a logarithmic function behaves is that it increases at a decreasing rate, which explains why it becomes integral in understanding scenarios like diminishing returns or measured decay processes.