Question: Find the derivative of the algebraic function $$H(v) = (v^6 - 5)(v^4 + 4)$$.
1. **State the problem:** Find the derivative of $$H(v) = (v^6 - 5)(v^4 + 4)$$.
2. **Formula used:** Use the product rule for derivatives: $$\frac{d}{dv}[f(v)g(v)] = f'(v)g(v) + f(v)g'(v)$$.
3. **Identify functions:** Let $$f(v) = v^6 - 5$$ and $$g(v) = v^4 + 4$$.
4. **Find derivatives:**
- $$f'(v) = 6v^5$$
- $$g'(v) = 4v^3$$
5. **Apply product rule:**
$$H'(v) = f'(v)g(v) + f(v)g'(v) = 6v^5(v^4 + 4) + (v^6 - 5)(4v^3)$$
6. **Expand terms:**
$$6v^5 \cdot v^4 = 6v^{9}$$
$$6v^5 \cdot 4 = 24v^{5}$$
$$v^6 \cdot 4v^3 = 4v^{9}$$
$$-5 \cdot 4v^3 = -20v^{3}$$
7. **Combine all terms:**
$$H'(v) = 6v^{9} + 24v^{5} + 4v^{9} - 20v^{3}$$
8. **Simplify like terms:**
$$6v^{9} + 4v^{9} = 10v^{9}$$
9. **Final derivative:**
$$H'(v) = 10v^{9} + 24v^{5} - 20v^{3}$$
10. **Answer:** This matches **Option D**.