Part 2/7:
An exponential function can be represented in the form ( f(x) = a^x ), where ( a ) is a constant. To find the derivative of such a function, we utilize the definition of the derivative. In previous discussions, it was established that the derivative, denoted as ( f'(x) ), can be calculated and results in a complex expression involving limits. For instance, using the definition, we find that:
[
f'(x) = a^x \cdot \lim_{h \to 0} \frac{a^h - 1}{h}
]
While this is mathematically rigorous, applying such complex formulas for practical uses like population growth becomes cumbersome.