Natural Exponents and Their Properties

Value of ( e^0 )

• ( e^0 = 1 ): Any non-zero number raised to the power of zero equals one.

Derivative of ( f(x) = e^x )

• The derivative ( f'(x) = e^x ): The function ( e^x ) is its own derivative, highlighting its unique property in calculus.

Property of ( e^{x+y} )

• ( e^{x+y} = e^x \cdot e^y ): This property illustrates the additive nature of the exponent when using the base ( e ).

Natural Exponentiation and Larger Exponents

• For natural exponentiation, larger exponents result in larger values: Exponential functions grow rapidly as the exponent increases, demonstrating the function's growth characteristics.

Natural Logarithm of ( e^x )

• The natural logarithm ( \ln(e^x) = x ): This relation shows that the natural logarithm function undoes the exponential function, returning the exponent itself.