Part 4/5:
This shows that the exponential function ( e^x ) is unique in that its rate of change at any point is equal to its value at that point.
Example with Base 2
Let’s consider a practical example where ( f(x) = 2^x ). Using our earlier steps, the derivative becomes:
[
f'(x) = 2^x \cdot \lim_{h \to 0} \frac{2^h - 1}{h}
]
It’s essential to remember that this limit approaches ( \ln(2) ), leading to the conclusion that the derivative of ( 2^x ) can be simplified to:
[
f'(x) = 2^x \cdot \ln(2)
]