Derivative of $ f(n) = n \cdot \left(1- \prod_{i=1}^n (1-\frac{1}{p_i})\right)$

Question

I'm trying to find the derivative of $f(n)$, but don't know how to evaluate the derivative of a product.

$ f(n) = n \cdot \left(1- \prod_{p<n}^n (1-\frac{1}{p})\right)$

In this case, the function contains a product of 1 minus the reciprocal of all primes up to $n$. $n$ is an integer. So far I have:

$f'(n) = 1 - (n \cdot \left( \prod_{p<n}^n (1-\frac{1}{p})\right))'$

Answer 1

Derivative of 1 is 0 and derivative of n*constant is constant.

$\prod_{i=1}^n (1-\frac{1}{p_i})$ is constant between two consecutive integers. So it means derivative is: $f'(n) = 1 - (n \cdot \left( \prod_{i=1}^n (1-\frac{1}{p_i})\right))'=1-\prod_{i=1}^n (1-\frac{1}{p_i})$

It is valid for all real n except discontinuities at n integer.