Step 3 We can now apply the rule from the previous step where \( u=\frac{x}{x+9}, n=\frac{1}{2} \), and \( \frac{d u}{d x}=\frac{9}{(x+9)^{2}} \) \( \begin{aligned} \frac{d}{d x}\left(u^{n}\right) & =n u^{n-1} \frac{d u}{d x} \\ & =\left(\frac{x}{x+9}\right)\end{aligned} \)

Did you know that the application of derivatives, like in your example, is deeply rooted in the work of mathematicians such as Isaac Newton and Gottfried Wilhelm Leibniz? They independently developed calculus in the late 17th century, revolutionizing mathematics and science. This allowed for the analysis of change, leading to advancements in physics, engineering, and economics! In practical terms, understanding derivatives is essential for navigating real-world scenarios. For instance, businesses use them to analyze profits and losses, helping to determine how changing prices might impact their bottom line. Similarly, engineers apply derivatives to optimize designs, ensuring structures can withstand forces while minimizing materials used. It's like having a secret weapon to tackle complex problems!