To find the differential of $y = \frac{s}{6 + 7s}$, we'll use the quotient rule, which is applied when differentiating a function of the form $\frac{f(s)}{g(s)}$.

The quotient rule is given by:

$\frac{d}{ds} \left( \frac{f(s)}{g(s)} \right) = \frac{f'(s) g(s) - f(s) g'(s)}{[g(s)]^2}$

Here:

• $f(s) = s$, and therefore $f'(s) = 1$
• $g(s) = 6 + 7s$, and therefore $g'(s) = 7$

Now, applying the quotient rule:

$\frac{dy}{ds} = \frac{(1)(6 + 7s) - (s)(7)}{(6 + 7s)^2}$

Simplify the numerator:

= \frac{6}{(6 + 7s)^2}$$ Thus, the derivative is: $$\frac{dy}{ds} = \frac{6}{(6 + 7s)^2}$$ Would you like further clarification or details on this? Here are five related questions that expand on this concept: 1. How would you differentiate $$y = \frac{3s}{5 + 2s^2}$$ using the quotient rule? 2. Can you find the derivative of $$y = \frac{1}{6 + 7s}$$ by a similar process? 3. How is the quotient rule related to the product rule in calculus? 4. What is the importance of simplifying the result after applying the quotient rule? 5. How would you apply the chain rule if the denominator were a more complicated function? **Tip:** When using the quotient rule, always simplify the result to make it more interpretable and easier to use in subsequent steps.