Differentiate the following w.r.t. x or t or u as the case may be:
11. `y= (x^2 + 2x +5)/( x^3 + 2x + 4)`

Answer

To differentiate the function \( y = \frac{x^2 + 2x + 5}{x^3 + 2x + 4} \) with respect to \( x \), we will use the quotient rule. The quotient rule states that if you have a function in the form \( y = \frac{u}{v} \), then the derivative \( \frac{dy}{dx} \) is given by:

\[ \frac{dy}{dx} = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \]

where:
- \( u = x^2 + 2x + 5 \)
- \( v = x^3 + 2x + 4 \)

### Step 1: Identify \( u \) and \( v \)
- \( u = x^2 + 2x + 5 \)
- \( v = x^3 + 2x + 4 \)

### Step 2: Differentiate \( u \) and \( v \)
- Differentiate \( u \):
\[ \frac{du}{dx} = 2x + 2 \]

- Differentiate \( v \):
\[ \frac{dv}{dx} = 3x^2 + 2 \]

### Step 3: Apply the Quotient Rule
Now, substitute \( u \), \( v \), \( \frac{du}{dx} \), and \( \frac{dv}{dx} \) into the quotient rule formula:

\[ \frac{dy}{dx} = \frac{(x^3 + 2x + 4)(2x + 2) - (x^2 + 2x + 5)(3x^2 + 2)}{(x^3 + 2x + 4)^2} \]

### Step 4: Simplify the Expression
Now we will simplify the numerator:

1. Expand \( (x^3 + 2x + 4)(2x + 2) \):
\[ = 2x^4 + 2x^3 + 4x + 4x^3 + 4x + 8 = 2x^4 + 6x^3 + 8x + 8 \]

2. Expand \( (x^2 + 2x + 5)(3x^2 + 2) \):
\[ = 3x^4 + 2x^2 + 6x^3 + 4x + 15x^2 + 10 = 3x^4 + 6x^3 + 17x^2 + 10 \]

3. Combine the results:
\[ \text{Numerator} = (2x^4 + 6x^3 + 8x + 8) - (3x^4 + 6x^3 + 17x^2 + 10) \]
\[ = 2x^4 - 3x^4 + 6x^3 - 6x^3 + 8x - 17x^2 + 8 - 10 \]
\[ = -x^4 - 17x^2 + 8x - 2 \]

### Final Result
Thus, the derivative of the function is:

\[ \frac{dy}{dx} = \frac{-x^4 - 17x^2 + 8x - 2}{(x^3 + 2x + 4)^2} \]