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.

Key Concepts

Product Rule

The Product Rule is a formula used to find the derivative of the product of two functions. If u(s) and v(s) are two differentiable functions, the derivative of their product is given by u'v + uv'. This rule is essential when dealing with functions that are multiplied together, allowing for the correct application of differentiation.

Quotient Rule

The Quotient Rule is used to differentiate a function that is the quotient of two other functions. If u(s) and v(s) are differentiable functions, the derivative of their quotient is given by (u'v - uv') / v². This rule is particularly important when the function is expressed as a fraction, ensuring accurate differentiation while considering the relationship between the numerator and denominator.

Simplification of Derivatives

After applying the Product or Quotient Rule, it is often necessary to simplify the resulting expression. This involves combining like terms, factoring, or reducing fractions to make the derivative easier to interpret and use. Simplification is a crucial step in calculus, as it helps clarify the behavior of the function and its rate of change.