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. \( \begin{array}{ll}\text { (c) } \\ h(w)=\sqrt[4]{w^{7}}+\frac{4}{w^{3}}\end{array} \)

We are given the function   h(w) = √[4]{w⁷} + 4/w³. Step 1. Rewrite the Function Using Exponents It’s easier to differentiate if we express the roots and reciprocals as exponents: • The fourth root of w⁷ can be written as w^(7/4), since √[4]{w⁷} = w^(7/4). • The term 4/w³ is equivalent to 4w^(–3). Thus, we have:   h(w) = w^(7/4) + 4w^(–3). Step 2. Differentiate Using the Power Rule and Sum Rule The rules we will use are:  • Sum Rule: The derivative of a sum is the sum of the derivatives.  • Power Rule: For any real number n, the derivative of wⁿ is n·w^(n–1).  • Constant Multiple Rule: The derivative of a constant times a function is the constant times the derivative of the function. Differentiate each term one at a time: A. Differentiate w^(7/4):  Using the Power Rule:   d/dw [w^(7/4)] = (7/4)·w^(7/4 – 1).  Compute the new exponent:   7/4 – 1 = (7/4 – 4/4) = 3/4.  Thus,   d/dw [w^(7/4)] = (7/4)·w^(3/4). B. Differentiate 4w^(–3):  Using the Constant Multiple Rule together with the Power Rule:   d/dw [4w^(–3)] = 4·d/dw [w^(–3)] = 4·(–3)w^(–3–1).  Compute the new exponent:   –3 – 1 = –4.  Thus,   d/dw [4w^(–3)] = –12w^(–4). Step 3. Combine the Results Now, apply the Sum Rule to add the derivatives of both terms:   h′(w) = (7/4)·w^(3/4) – 12w^(–4). This is the derivative of the function h(w). Summary of Derivative Rules Used: 1. Sum Rule: d/dw [f(w) + g(w)] = f′(w) + g′(w). 2. Power Rule: d/dw [wⁿ] = n·w^(n–1) (applied to both terms). 3. Constant Multiple Rule: d/dw [c·f(w)] = c·f′(w). Thus, the final answer is:   h′(w) = (7/4)w^(3/4) – 12w^(–4).