Differentiate the following w.r.t. x :
`sqrt(((x-3)(x^(2)+4))/(3x^(2)+4x+5))`

Answer

To differentiate the function \( y = \sqrt{\frac{(x-3)(x^2+4)}{3x^2+4x+5}} \) with respect to \( x \), we will use the chain rule and the quotient rule. Here’s a step-by-step solution:

### Step 1: Rewrite the function
Let:
\[ y = \sqrt{\frac{(x-3)(x^2+4)}{3x^2+4x+5}} \]
We can rewrite this as:
\[ y = \left(\frac{(x-3)(x^2+4)}{3x^2+4x+5}\right)^{1/2} \]

### Step 2: Differentiate using the chain rule
Using the chain rule, the derivative \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = \frac{1}{2} \left(\frac{(x-3)(x^2+4)}{3x^2+4x+5}\right)^{-1/2} \cdot \frac{d}{dx}\left(\frac{(x-3)(x^2+4)}{3x^2+4x+5}\right) \]

### Step 3: Differentiate the inner function using the quotient rule
Let:
\[ u = (x-3)(x^2+4) \quad \text{and} \quad v = 3x^2 + 4x + 5 \]
Using the quotient rule:
\[ \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \]

### Step 4: Find \( \frac{du}{dx} \) and \( \frac{dv}{dx} \)
1. **Differentiate \( u \)**:
\[ u = (x-3)(x^2+4) \]
Using the product rule:
\[ \frac{du}{dx} = (x^2+4) \cdot 1 + (x-3) \cdot 2x = x^2 + 4 + 2x^2 - 6x = 3x^2 - 6x + 4 \]

2. **Differentiate \( v \)**:
\[ v = 3x^2 + 4x + 5 \]
\[ \frac{dv}{dx} = 6x + 4 \]

### Step 5: Substitute back into the quotient rule
Now substituting \( u \), \( v \), \( \frac{du}{dx} \), and \( \frac{dv}{dx} \) into the quotient rule:
\[ \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{(3x^2 + 4x + 5)(3x^2 - 6x + 4) - (x-3)(x^2 + 4)(6x + 4)}{(3x^2 + 4x + 5)^2} \]

### Step 6: Combine everything
Now substituting this result back into the derivative of \( y \):
\[ \frac{dy}{dx} = \frac{1}{2} \left(\frac{(x-3)(x^2+4)}{3x^2+4x+5}\right)^{-1/2} \cdot \frac{(3x^2 + 4x + 5)(3x^2 - 6x + 4) - (x-3)(x^2 + 4)(6x + 4)}{(3x^2 + 4x + 5)^2} \]

### Step 7: Final expression
Thus, the final expression for the derivative is:
\[ \frac{dy}{dx} = \frac{1}{2\sqrt{\frac{(x-3)(x^2+4)}{3x^2+4x+5}}} \cdot \frac{(3x^2 + 4x + 5)(3x^2 - 6x + 4) - (x-3)(x^2 + 4)(6x + 4)}{(3x^2 + 4x + 5)^2} \]