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

To solve the derivative of the quotient \( \frac{u}{v} \) correctly, make sure to clarify what \( u \) and \( v \) represent. If \( y = \frac{2x}{\sqrt{x-5}} \), then let’s identify \( u = 2x \) and \( v = \sqrt{x-5} \). The quotient rule states that the derivative is given by: \[ \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \] By applying the rule, we differentiate \( u \) and \( v \) to find \( \frac{du}{dx} = 2 \) and \( \frac{dv}{dx} = \frac{1}{2\sqrt{x-5}} \). This can then be substituted into the formula to compute the derivative of \( y \). Now, let's address two areas related to this topic. When dealing with differentials and derivatives, knowing the historical context of calculus enhances your understanding! The quotient rule, like many calculus concepts, traces back to the work of Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century. Their exploration of rates of change laid the groundwork for modern-day calculus and its applications across various fields, making them the superstars of mathematical history! In practical applications, mastering the quotient rule is essential in fields ranging from physics to economics. For example, when calculating efficiency ratios, velocity in physics, or even in optimizing revenue in business, you’ll find yourself using this technique. Understanding how to navigate these calculations not only builds mathematical skills but can also lead to insights crucial for decision-making in professional scenarios.