Part 4/7:
This relationship leads us to recognize that if ( f'(x) ) is defined using ( e^x ), then its derivative is simply ( f'(x) = e^x ). This self-referential property makes ( e ) immensely useful in calculus, as it allows for easy evaluation at any point on the curve. The numerical value of ( e ) approximates to ( 2.71828 ), but its significance extends far beyond its numerical representation.
Comparisons with Other Exponential Functions
To understand ( e ) better, we can compare its derivative to those of other exponential functions. For instance, consider the functions ( f_1(x) = 2^x ) and ( f_2(x) = 3^x ). The derivative for these can similarly be expressed, showcasing that: