10. Demonstrate and explain how to find the derivative of the following functions. Be sure to explicitly denote which derivative rules (scalar multiple, sum/difference, quotient, etc.) you are using in your work. (b) \[ f(y)=-\frac{3 y^{2}+5 y-3}{y^{3}} \]

We wish to differentiate   f(y) = – (3y² + 5y – 3) / y³. One convenient strategy is to simplify f(y) first and then differentiate term by term using the power rule. We will also note along the way which rules are being used (constant multiple rule, sum/difference rule, and power rule). Step 1. Simplify the Function Recall that division by y³ allows us to rewrite each term:   (3y²)/y³ = 3y²⁻³ = 3y⁻¹,   (5y)/y³ = 5y¹⁻³ = 5y⁻²,   (–3)/y³ = –3y⁻³. Thus,   f(y) = –[3y² + 5y – 3] / y³      = –(3y² / y³ + 5y / y³ – 3/y³)      = –(3y⁻¹ + 5y⁻² – 3y⁻³). Distribute the negative sign:   f(y) = –3y⁻¹ – 5y⁻² + 3y⁻³. Step 2. Differentiate Term by Term We now differentiate each term using the power rule. Recall that the power rule states that d/dy (yⁿ) = n yⁿ⁻¹, and we also apply the constant multiple rule. 1. Differentiate –3y⁻¹:   d/dy [–3y⁻¹] = –3 · d/dy (y⁻¹)           = –3 · (–1)y⁻²                     [Power rule]           = 3y⁻². 2. Differentiate –5y⁻²:   d/dy [–5y⁻²] = –5 · d/dy (y⁻²)           = –5 · (–2)y⁻³                     [Power rule]           = 10y⁻³. 3. Differentiate 3y⁻³:   d/dy [3y⁻³] = 3 · d/dy (y⁻³)           = 3 · (–3)y⁻⁴                     [Power rule]           = –9y⁻⁴. Now, summing these results using the sum rule we obtain:   f′(y) = 3y⁻² + 10y⁻³ – 9y⁻⁴. Step 3. Express the Derivative in Fraction Form (Optional) Since y⁻² = 1/y², y⁻³ = 1/y³, and y⁻⁴ = 1/y⁴, we can rewrite the derivative as:   f′(y) = 3/y² + 10/y³ – 9/y⁴. For a common denominator, note that:   3y⁻² = 3y²/y⁴, 10y⁻³ = 10y/y⁴, –9y⁻⁴ = –9/y⁴. Thus,   f′(y) = (3y² + 10y – 9) / y⁴. Conclusion: We have used the constant multiple rule, the sum/difference rule, and the power rule to differentiate f(y). The final answer is:   f′(y) = 3y⁻² + 10y⁻³ – 9y⁻⁴  or   f′(y) = (3y² + 10y – 9) / y⁴.