Find the derivatives of the following :
(i) `(x^(3)3x^(2)+4)(4x^(5)+x^(2)-1)` (ii) `(9x^(2))/(x-3)`

Options

A. `sin 74^(@)`
B. `cos 106^(@)`
C. `sin 15^(@)`
D. `cos 75^(@)`

Explanation

To find the derivatives of the given functions, we will use the product rule and the quotient rule of differentiation.

### Part (i): Find the derivative of \( (x^3 + 4)(4x^5 + x^2 - 1) \)

1. **Identify the functions**: Let \( u = x^3 + 4 \) and \( v = 4x^5 + x^2 - 1 \).

2. **Differentiate \( u \) and \( v \)**:
- \( u' = \frac{d}{dx}(x^3 + 4) = 3x^2 \)
- \( v' = \frac{d}{dx}(4x^5 + x^2 - 1) = 20x^4 + 2x \)

3. **Apply the product rule**: The product rule states that \( \frac{d}{dx}(uv) = u'v + uv' \).
- Thus, the derivative \( \frac{d}{dx}[(x^3 + 4)(4x^5 + x^2 - 1)] = (3x^2)(4x^5 + x^2 - 1) + (x^3 + 4)(20x^4 + 2x) \).

4. **Expand the expression**:
- First term: \( 3x^2(4x^5 + x^2 - 1) = 12x^7 + 3x^4 - 3x^2 \)
- Second term: \( (x^3 + 4)(20x^4 + 2x) = x^3(20x^4 + 2x) + 4(20x^4 + 2x) = 20x^7 + 2x^4 + 80x^4 + 8x = 20x^7 + 82x^4 + 8x \)

5. **Combine like terms**:
- Combine \( 12x^7 + 20x^7 = 32x^7 \)
- Combine \( 3x^4 + 82x^4 = 85x^4 \)
- The final derivative is \( 32x^7 + 85x^4 + 5x^2 \).

### Final answer for part (i):
\[ \frac{d}{dx}[(x^3 + 4)(4x^5 + x^2 - 1)] = 32x^7 + 85x^4 + 5x^2 \]

---

### Part (ii): Find the derivative of \( \frac{9x^2}{x - 3} \)

1. **Identify the functions**: Let \( u = 9x^2 \) and \( v = x - 3 \).

2. **Differentiate \( u \) and \( v \)**:
- \( u' = \frac{d}{dx}(9x^2) = 18x \)
- \( v' = \frac{d}{dx}(x - 3) = 1 \)

3. **Apply the quotient rule**: The quotient rule states that \( \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v u' - u v'}{v^2} \).
- Thus, the derivative is \( \frac{(x - 3)(18x) - (9x^2)(1)}{(x - 3)^2} \).

4. **Expand the expression**:
- Numerator: \( (x - 3)(18x) - 9x^2 = 18x^2 - 54x - 9x^2 = 9x^2 - 54x \)

5. **Final expression**:
- The derivative is \( \frac{9x^2 - 54x}{(x - 3)^2} \).

### Final answer for part (ii):
\[ \frac{d}{dx}\left(\frac{9x^2}{x - 3}\right) = \frac{9x^2 - 54x}{(x - 3)^2} \]

---