(i) `(2x+3) (3x-5)` (ii) `x(1+x)^(3) ` (iii) ` (sqrtx + 1/x) (x -1/sqrtx)`
(iv) ` (x-1/x)^(2)` (v) ` (x^(2) + 1/x^(2))^(3)` (vi) `(2x^(2) +5x-1) (x-3)`

Answer

Let's solve the given problems step by step.

### (i) Differentiate `(2x + 3)(3x - 5)`

1. **Identify the functions**: Let \( u = 2x + 3 \) and \( v = 3x - 5 \).
2. **Apply the product rule**: The product rule states that \( \frac{d}{dx}(uv) = u \frac{dv}{dx} + v \frac{du}{dx} \).
3. **Find derivatives**:
- \( \frac{du}{dx} = 2 \)
- \( \frac{dv}{dx} = 3 \)
4. **Substitute into the product rule**:
\[ \frac{d}{dx}((2x + 3)(3x - 5)) = (2x + 3)(3) + (3x - 5)(2) \]
5. **Simplify**:
\[ = 6x + 9 + 6x - 10 = 12x - 1 \]

### (ii) Differentiate `x(1 + x)^3`

1. **Identify the functions**: Let \( u = x \) and \( v = (1 + x)^3 \).
2. **Apply the product rule**:
\[ \frac{d}{dx}(uv) = u \frac{dv}{dx} + v \frac{du}{dx} \]
3. **Find derivatives**:
- \( \frac{du}{dx} = 1 \)
- \( \frac{dv}{dx} = 3(1 + x)^2 \cdot 1 = 3(1 + x)^2 \)
4. **Substitute into the product rule**:
\[ \frac{d}{dx}(x(1 + x)^3) = x(3(1 + x)^2) + (1 + x)^3(1) \]
5. **Simplify**:
\[ = 3x(1 + x)^2 + (1 + x)^3 \]

### (iii) Differentiate `(√x + 1/x)(x - 1/√x)`

1. **Identify the functions**: Let \( u = \sqrt{x} + \frac{1}{x} \) and \( v = x - \frac{1}{\sqrt{x}} \).
2. **Apply the product rule**:
\[ \frac{d}{dx}(uv) = u \frac{dv}{dx} + v \frac{du}{dx} \]
3. **Find derivatives**:
- \( \frac{du}{dx} = \frac{1}{2\sqrt{x}} - \frac{1}{x^2} \)
- \( \frac{dv}{dx} = 1 + \frac{1}{2}x^{-3/2} \)
4. **Substitute into the product rule**:
\[ = \left(\sqrt{x} + \frac{1}{x}\right)\left(1 + \frac{1}{2\sqrt{x}}\right) + \left(x - \frac{1}{\sqrt{x}}\right)\left(\frac{1}{2\sqrt{x}} - \frac{1}{x^2}\right) \]

### (iv) Differentiate `(x - 1/x)^2`

1. **Let \( y = (x - \frac{1}{x})^2 \)**.
2. **Use the chain rule**:
\[ \frac{dy}{dx} = 2(x - \frac{1}{x})\left(1 + \frac{1}{x^2}\right) \]
3. **Simplify**:
\[ = 2(x - \frac{1}{x})(1 + \frac{1}{x^2}) \]

### (v) Differentiate `(x^2 + 1/x^2)^3`

1. **Let \( y = (x^2 + \frac{1}{x^2})^3 \)**.
2. **Use the chain rule**:
\[ \frac{dy}{dx} = 3(x^2 + \frac{1}{x^2})^2\left(2x - \frac{2}{x^3}\right) \]

### (vi) Differentiate `(2x^2 + 5x - 1)(x - 3)`

1. **Identify the functions**: Let \( u = 2x^2 + 5x - 1 \) and \( v = x - 3 \).
2. **Apply the product rule**:
\[ \frac{d}{dx}(uv) = u \frac{dv}{dx} + v \frac{du}{dx} \]
3. **Find derivatives**:
- \( \frac{du}{dx} = 4x + 5 \)
- \( \frac{dv}{dx} = 1 \)
4. **Substitute into the product rule**:
\[ = (2x^2 + 5x - 1)(1) + (x - 3)(4x + 5) \]