To differentiate the function $G(s) = \frac{5s^2}{7e^s + s}$, we will apply the quotient rule for differentiation, which is given by:

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

Here, we identify:

• $f(s) = 5s^2$
• $g(s) = 7e^s + s$

Step 1: Differentiate $f(s) = 5s^2$

$f'(s) = 10s$

Step 2: Differentiate $g(s) = 7e^s + s$

$g'(s) = 7e^s + 1$

Step 3: Apply the quotient rule

$G'(s) = \frac{(10s)(7e^s + s) - (5s^2)(7e^s + 1)}{(7e^s + s)^2}$

Now, let's simplify the numerator:

1. Expand the first term: $(10s)(7e^s + s) = 70se^s + 10s^2$
2. Expand the second term: $(5s^2)(7e^s + 1) = 35s^2 e^s + 5s^2$

So the numerator becomes: $70se^s + 10s^2 - 35s^2 e^s - 5s^2$

Now simplify the numerator: $(70se^s - 35s^2 e^s) + (10s^2 - 5s^2) = 35se^s(2 - s) + 5s^2$

Thus, the derivative is: $G'(s) = \frac{35se^s(2 - s) + 5s^2}{(7e^s + s)^2}$

This is the derivative of the given function $G(s)$.

Would you like more details or clarifications on any step?

Here are 5 related questions:

1. How does the quotient rule apply in different contexts?
2. What are the key differences between the quotient and product rules in differentiation?
3. How do exponential terms affect differentiation processes?
4. Can this function be solved using logarithmic differentiation?
5. How does the chain rule combine with the quotient rule in more complex functions?

Tip: Always check for common factors after applying the quotient rule to simplify your final result.