Because with that particular case a=e the exponent function a^x is its own derivative, and because of uniqueness this means it is the only function that remains unchanged when differentiated and integrated.
d/dx e^x = e^x , and, int e^x \ dx = e^x " "(+c)
e is an irrational number (like pi or sqrt(2)), and like pi the number crops up in many natural circumstances (e.g. radioactive decay, temperature cooling, compound interest, probability etc.)
It is also related to pi, and i the imaginary number by Euler's famous identity:
e^(ipi) + 1 -= 0
e is also the base used for "Natural" Logarithms,denoted by lnx, where:
lnx=a <=> x=e^a
And the Natural Logarithm has the property of being the area under the curve y=1/x, as
int_1^x \ 1/t \ dt = lnx => int_1^e \ 1/t \ dt = 1
e^x can be expanded as a Power Series (using Taylor's Theorem) to give;
e^x = 1+x+x^2/(2!)+x^3/(3!)+x^4/(4!) + ... = sum_(n=0)^oo x^n/(n!)
which is convergent for all real values of x, And so e itself can be written as the infinite sum:
e = 1+1+1/(2!)+1/(3!)+1/(4!) + ... = sum_(n=0)^oo 1/(n!)
and that e is the value of the limit:
lim_(n rarr oo) (1 + 1/n)^n
Euler gave an approximation for e to 18 decimal places,
e = 2.718281828459045235
