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. \( \begin{array}{ll}\text { (d) } \\ g(t)=7 t^{5}-6 t^{2}-t-4\end{array} \)

We want to differentiate g(t) = 7t⁵ – 6t² – t – 4. We will use several derivative rules: 1. Power Rule: d/dt(tⁿ) = n·tⁿ⁻¹. 2. Constant Multiple Rule: d/dt[c · f(t)] = c · f '(t). 3. Sum (and Difference) Rule: The derivative of a sum/difference is the sum/difference of the derivatives. 4. Constant Rule: d/dt[c] = 0. Step-by-step: 1. Differentiate 7t⁵:   Using the Constant Multiple and Power Rules:   d/dt[7t⁵] = 7 · d/dt[t⁵] = 7 · (5t⁴) = 35t⁴. 2. Differentiate –6t²:   d/dt[–6t²] = –6 · d/dt[t²] = –6 · (2t) = –12t. 3. Differentiate –t:   Note that –t is the same as –1·t:   d/dt[–t] = –1. 4. Differentiate –4:   Since –4 is a constant, its derivative is 0. Putting it all together using the Sum/Difference Rule: g '(t) = 35t⁴ – 12t – 1 + 0 So, the derivative is:   g '(t) = 35t⁴ – 12t – 1.