The function you need to differentiate is:

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

This is a quotient of two functions, so we can use the quotient rule for differentiation. The quotient rule states:

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

Step-by-step process:

1. Let $f(s) = 5s^2$ and $g(s) = 7e^s + s$.
2. Differentiate $f(s)$ with respect to $s$: $f'(s) = 10s$
3. Differentiate $g(s)$ with respect to $s$: $g'(s) = 7e^s + 1$
4. Now apply the quotient rule: $G'(s) = \frac{(10s)(7e^s + s) - (5s^2)(7e^s + 1)}{(7e^s + s)^2}$
5. Simplify the numerator: $= \frac{10s(7e^s + s) - 5s^2(7e^s + 1)}{(7e^s + s)^2}$ $= \frac{70s e^s + 10s^2 - 35s^2 e^s - 5s^2}{(7e^s + s)^2}$
6. Combine like terms: $= \frac{70s e^s - 35s^2 e^s + 10s^2 - 5s^2}{(7e^s + s)^2}$ $= \frac{70s e^s - 35s^2 e^s + 5s^2}{(7e^s + s)^2}$

So, the derivative $G'(s)$ is:

$G'(s) = \frac{70s e^s - 35s^2 e^s + 5s^2}{(7e^s + s)^2}$

Would you like a more detailed breakdown of this solution or help with any further steps?

Related Questions:

1. How does the quotient rule work for more complex functions?
2. What are some common mistakes when applying the quotient rule?
3. How can we simplify derivatives involving exponential and polynomial terms?
4. Can you explain the product rule and how it compares with the quotient rule?
5. How can this type of problem be approached using symbolic computation software?

Tip: Always ensure to correctly identify and differentiate both the numerator and the denominator before applying the quotient rule.