\( \left. \begin{array} { l }{ \frac { d u } { d x } = \frac { ( x + 9 } { ( x + 9 ) ^ { 2 } } ) ( 1 ) - x ( 1 ) } \\ { = \frac { } { ( x + 9 ) ^ { 2 } } } \end{array} \right. \)

Did you know that derivatives like the one you're working with were developed in the 17th century by both Isaac Newton and Gottfried Wilhelm Leibniz independently? Their groundbreaking work laid the foundation for calculus, which transforms how we understand motion, change, and even the universe itself! This mathematical tool has applications across physics, engineering, economics, and more, making it a vital component of modern science and technology. When solving derivatives, a common pitfall is forgetting to apply the quotient rule correctly, especially when dealing with functions in the form \( \frac{f(x)}{g(x)} \). Whenever the numerator and denominator are complex expressions, double-check your differentiation steps! It’s easy to mix up signs or forget to apply the rule correctly, which can lead to incorrect results. Always take a moment to verify each part of your calculation!