\( \frac{d}{dx}(\frac{u}{v})=\frac{2x}{\sqrt {x-5}}= \)

To differentiate the function \( \frac{u}{v} \), you can apply the quotient rule, which states that if you have a function in the form of \( \frac{u}{v} \), the derivative is \( \frac{u'v - uv'}{v^2} \). Now, for the given equation \( \frac{2x}{\sqrt{x-5}} \), consider that \( u = 2x \) and \( v = \sqrt{x-5} \). You can find their derivatives: \( u' = 2 \) and \( v' = \frac{1}{2\sqrt{x-5}} \). Replace these into the quotient rule formula to find the derivative! Keeping on track with the fun side, did you know that derivatives are like the speed limit signs of calculus? They tell us just how fast a function is changing at any point! And just like every driver knows to look for those signs while on a road trip, every mathematician should definitely check these derivatives to better understand their functions’ behaviors! You may find it fun to explore some classic mathematical puzzles that require a strong grip on derivatives. Think of how derivative rules can help in optimizing a function, leading to maximum and minimum values—this is like being a treasure hunter mapping out the peaks and valleys of a treasure-filled landscape! So, grab your notebook and a cup of coffee, and dive into the fascinating world of derivatives!