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An exponential function is a type of mathematical function that involves an exponent. It has the form of \( f(x) = a^x \), where \( a \) is a constant, and \( x \) is the variable.
An easy example of an exponential function is \( 2^x \), where each integer increase in \( x \) results in a doubling of the value of \( f(x) \).
Exponential functions are important in various fields such as biology (population growth), finance (compound interest), and physics (radioactive decay).
They are unique because their growth rate is proportional to their current value, leading to a rapid increase or decrease.

Differentiation is a fundamental concept in calculus. It involves finding the 'derivative' of a function, which tells us how the function changes at any point.
One of the basic derivative rules is the power rule, \( \frac{d}{dx}(x^n) = nx^{n-1} \). However, exponential functions like \( a^x \) require a different approach.
The general rule for differentiating an exponential function \( a^x \) is: \( \frac{d}{dx}(a^x) = a^x \times \ln(a) \).
In our example, \( f(x) = 100^x \), applying this rule gives us the derivative \( 100^x \times \ln(100) \). This rule simplifies the process and helps us understand how rapid growth or decay occurs in exponential functions.

The natural logarithm, denoted as \( \ln \), is a logarithm to the base \( e \), where \( e \approx 2.71828 \). It is widely used in calculus and mathematical analysis.
One key property of the natural logarithm is that the derivative of the natural logarithm function \( \frac{d}{dx}(\text{ln}(x)) = \frac{1}{x} \).
Moreover, the natural logarithm appears when differentiating exponential functions like \( a^x \). For example, when differentiating \( 100^x \), we get \( 100^x \times \ln(100) \), where \( \ln(100) \) is constant and simplifies the expression.
This interrelationship between exponential functions and natural logarithms is crucial for solving many mathematical problems involving growth and decay processes.