7–14. Find the derivative the following ways:
a. Using the Product Rule (Exercises 7–10) or the Quotient Rule (Exercises 11–14). Simplify your result.
g(s) = 4s³ - 8s² +4s / 4s

Verified step by step guidance

1. Step 1: Identify the function g(s) = \frac{4s^3 - 8s^2 + 4s}{4s}. This is a rational function, so we will use the Quotient Rule to find its derivative.
2. Step 2: Recall the Quotient Rule, which states that if you have a function h(s) = \frac{u(s)}{v(s)}, then the derivative h'(s) is given by \frac{u'(s)v(s) - u(s)v'(s)}{(v(s))^2}.
3. Step 3: Assign u(s) = 4s^3 - 8s^2 + 4s and v(s) = 4s. Calculate the derivatives: u'(s) = \frac{d}{ds}(4s^3 - 8s^2 + 4s) and v'(s) = \frac{d}{ds}(4s).
4. Step 4: Compute u'(s) = 12s^2 - 16s + 4 and v'(s) = 4. Substitute these into the Quotient Rule formula: g'(s) = \frac{(12s^2 - 16s + 4)(4s) - (4s^3 - 8s^2 + 4s)(4)}{(4s)^2}.
5. Step 5: Simplify the expression for g'(s) by expanding the terms in the numerator and then combining like terms. Finally, simplify the entire expression by dividing each term by (4s)^2.