Find `(dy)/(dx)` for the following :
(i) `y=x^(7//2)` (ii) `y=x^(-3)`
`(iii) y=x` (iv) `y=x^(5)+x^(3)+4x^(1//2)+7`
(v) `y=5x^(4)+6x^(3//2)+9x` (vi) `y=ax^(2)+bx+C`
`(Vii) `y=3x^(5)-3x-1/x`

Explanation

To find \(\frac{dy}{dx}\) for the given functions, we will use the power rule of differentiation, which states that if \(y = x^n\), then \(\frac{dy}{dx} = n \cdot x^{n-1}\).

### Step-by-Step Solutions:

1. **For \(y = x^{7/2}\)**:
\[ \frac{dy}{dx} = \frac{7}{2} x^{7/2 - 1} = \frac{7}{2} x^{5/2} \]

2. **For \(y = x^{-3}\)**:
\[ \frac{dy}{dx} = -3 x^{-3 - 1} = -3 x^{-4} \]

3. **For \(y = x\)**:
\[ \frac{dy}{dx} = 1 \cdot x^{1 - 1} = 1 \cdot x^0 = 1 \]

4. **For \(y = x^5 + x^3 + 4x^{1/2} + 7\)**:
\[ \frac{dy}{dx} = 5x^{4} + 3x^{2} + 4 \cdot \frac{1}{2} x^{-1/2} + 0 = 5x^{4} + 3x^{2} + 2x^{-1/2} \]

5. **For \(y = 5x^4 + 6x^{3/2} + 9x\)**:
\[ \frac{dy}{dx} = 20x^{3} + 6 \cdot \frac{3}{2} x^{1/2} + 9 = 20x^{3} + 9x^{1/2} + 9 \]

6. **For \(y = ax^2 + bx + C\)**:
\[ \frac{dy}{dx} = 2ax + b \]

7. **For \(y = 3x^5 - 3x - \frac{1}{x}\)**:
\[ \frac{dy}{dx} = 15x^{4} - 3 + \frac{1}{x^2} = 15x^{4} - 3 + x^{-2} \]

### Summary of Results:
- (i) \(\frac{dy}{dx} = \frac{7}{2} x^{5/2}\)
- (ii) \(\frac{dy}{dx} = -3 x^{-4}\)
- (iii) \(\frac{dy}{dx} = 1\)
- (iv) \(\frac{dy}{dx} = 5x^{4} + 3x^{2} + 2x^{-1/2}\)
- (v) \(\frac{dy}{dx} = 20x^{3} + 9x^{1/2} + 9\)
- (vi) \(\frac{dy}{dx} = 2ax + b\)
- (vii) \(\frac{dy}{dx} = 15x^{4} - 3 + x^{-2}\)