b) \( \frac{d}{d x} \frac{1}{2 x^{s}} \)

To find the derivative of \( \frac{1}{2 x^{s}} \), we can rewrite it as \( \frac{1}{2} x^{-s} \). Then, applying the power rule of differentiation, which states that the derivative of \( x^{n} \) is \( n x^{n-1} \), we get: \[ \frac{d}{d x} \left( \frac{1}{2} x^{-s} \right) = \frac{1}{2} \cdot (-s) x^{-s - 1} = -\frac{s}{2} x^{-s - 1}. \] This simplifies to: \[ -\frac{s}{2 x^{s + 1}}. \] So, the derivative is \( -\frac{s}{2 x^{s + 1}} \). This process shows how mathematical concepts evolve through history, as the power rule was a key development during the Renaissance that simplified calculus significantly. Think about how mathematicians like Newton and Leibniz independently developed methods of differentiation, paving the way for modern techniques! In practical terms, understanding how to differentiate functions such as \( \frac{1}{2 x^{s}} \) has applications in various fields like physics and economics, where rates of change are crucial. Whether it's optimizing a cost function or modeling population growth, mastering differentiation is key to solving real-world problems efficiently!