Obtain the different coefficient of the following :
(i) `(x-1)(2x+5)`
(ii) `(1)/(2x+1)`
(iii) `(3x+4)/(4x+5)`
(iv) `(x^(2))/(x^(3)+1)`

Explanation

To find the differential coefficients of the given functions, we will apply the rules of differentiation step by step.

### (i) For the function \( y = (x - 1)(2x + 5) \)

1. **Identify the functions**: Let \( u = x - 1 \) and \( v = 2x + 5 \).
2. **Differentiate**:
- \( \frac{du}{dx} = 1 \)
- \( \frac{dv}{dx} = 2 \)
3. **Apply the product rule**:
\[ \frac{dy}{dx} = u \frac{dv}{dx} + v \frac{du}{dx} = (x - 1)(2) + (2x + 5)(1) \]
4. **Simplify**:
\[ \frac{dy}{dx} = 2(x - 1) + (2x + 5) = 2x - 2 + 2x + 5 = 4x + 3 \]

### (ii) For the function \( y = \frac{1}{2x + 1} \)

1. **Rewrite the function**: \( y = (2x + 1)^{-1} \).
2. **Differentiate using the chain rule**:
\[ \frac{dy}{dx} = -1(2x + 1)^{-2} \cdot \frac{d(2x + 1)}{dx} = -1(2x + 1)^{-2} \cdot 2 \]
3. **Simplify**:
\[ \frac{dy}{dx} = -\frac{2}{(2x + 1)^2} \]

### (iii) For the function \( y = \frac{3x + 4}{4x + 5} \)

1. **Identify the functions**: Let \( u = 3x + 4 \) and \( v = 4x + 5 \).
2. **Differentiate**:
- \( \frac{du}{dx} = 3 \)
- \( \frac{dv}{dx} = 4 \)
3. **Apply the quotient rule**:
\[ \frac{dy}{dx} = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} = \frac{(4x + 5)(3) - (3x + 4)(4)}{(4x + 5)^2} \]
4. **Simplify**:
\[ = \frac{12x + 15 - (12x + 16)}{(4x + 5)^2} = \frac{-1}{(4x + 5)^2} \]

### (iv) For the function \( y = \frac{x^2}{x^3 + 1} \)

1. **Identify the functions**: Let \( u = x^2 \) and \( v = x^3 + 1 \).
2. **Differentiate**:
- \( \frac{du}{dx} = 2x \)
- \( \frac{dv}{dx} = 3x^2 \)
3. **Apply the quotient rule**:
\[ \frac{dy}{dx} = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} = \frac{(x^3 + 1)(2x) - (x^2)(3x^2)}{(x^3 + 1)^2} \]
4. **Simplify**:
\[ = \frac{2x^4 + 2x - 3x^4}{(x^3 + 1)^2} = \frac{-x^4 + 2x}{(x^3 + 1)^2} \]

### Summary of Results
1. \( \frac{dy}{dx} = 4x + 3 \) for \( (x - 1)(2x + 5) \)
2. \( \frac{dy}{dx} = -\frac{2}{(2x + 1)^2} \) for \( \frac{1}{2x + 1} \)
3. \( \frac{dy}{dx} = -\frac{1}{(4x + 5)^2} \) for \( \frac{3x + 4}{4x + 5} \)
4. \( \frac{dy}{dx} = \frac{-x^4 + 2x}{(x^3 + 1)^2} \) for \( \frac{x^2}{x^3 + 1} \)