Part 2/5:
An exponential function is generally expressed as ( f(x) = a^x ), where ( a ) is a constant and ( x ) is a variable. It is essential to differentiate this from power functions like ( x^a ). In previous discussions, notably in part one of this series, a more complex method based on the definition of a derivative was used. However, the objective is to simplify the process using the chain rule, making it considerably easier to comprehend and apply.
Deriving the Exponential Function
To derive the function ( f(x) = a^x ), we start from the limit definition of derivative:
[
f'(x) = \lim_{h \to 0} \frac{a^{x+h} - a^x}{h}
]