Derivative and Integral of $\pi(x)$ (primes before $x$)? What's the derivative and indefinite integral of $\pi(x)$, which gives the number of primes before $x$? I know there's no answer in terms of elementary functions, but is there an answer in terms of other functions or series?

The derivative of π(x) would be 0 except when x is a prime, when it wouldn't exist since  π(x) takes a step at prime values of x.
