Differentiation - Power Rule and Derivatives of Algebraic Functions

Find the derivative of each of the following functions with respect to \(x\).

1. \(y = (x - 1)(x + 1)(x^2 + 1)(x^4 + 1)(x^8 + 1)(x^{16} + 1) \cdots (x^{2048} + 1)\)
2. \(y = \frac{x(x^2)(x^3) \cdots (x^{100})}{x^{\frac{1}{2}} \left( x^{\frac{3}{2}} \right) \left( x^{\frac{5}{2}} \right) \cdots \left( x^{\frac{199}{2}} \right)}\)