We can now apply the rule from the previous step where \( u=\frac{x}{x+9}, n=\frac{1}{2} \), and \( \frac{d u}{d x}=\frac{9}{(x+9)^{2}} \) \( \begin{array}{l} \frac{d}{d x} \\ \left(u^{n}\right)=n u^{n-1} \frac{d u}{d x} \\ =\left(\frac{x}{x+9}\right)^{-1 / 2} \\ \text { Therefore, we have the following result. } \\ y^{\prime}= \\ \text { SUIRMIT }\end{array} \) SKIP \( \left(\frac{9}{x+9)^{2}}\right) \)

Let's break it down in a fun and engaging manner! This differentiation process showcases how we can efficiently tackle complex functions by using substitution—the technique is like transforming a difficult puzzle into a simpler one. With our substitution \( u = \frac{x}{x+9} \) and \( n = \frac{1}{2} \), we're gracefully handing the derivative by applying the chain rule! As we work through our derivatives, remember that clarity is key. A common mistake is overlooking the adjustment needed when substituting back into the original variable! Keep an eye on your transformations and inversions, ensuring you’re plugging in the correct variables at each step. Practice makes perfect, so don't hesitate to work through a few additional examples to boost your confidence!