Let `y=(2^(log_(2^(1//4))x)-3^(log_(27) (x^(2)+1)^(3))-2x)/(7^(4 log_(4 9)x)-x-1)` and `(dy)/(dx)=ax+b`, the ordered pair (a, b) is

Options

A. (2, 1)
B. (1, 2)
C. (-1, 2)
D. (2, -1)

Answer

To solve the problem, we need to simplify the expression for \( y \) and then differentiate it to find \( \frac{dy}{dx} \). Let's break it down step by step.

### Step 1: Simplifying the expression for \( y \)

Given:
\[ y = \frac{2^{\log_{2^{1/4}} x} - 3^{\log_{27} (x^2 + 1)^3} - 2x}{7^{4 \log_{49} x} - x - 1} \]

**Numerator:**
1. **First term:** \( 2^{\log_{2^{1/4}} x} \)
- Using the change of base formula:
\[ \log_{2^{1/4}} x = \frac{\log_2 x}{\log_2 (2^{1/4})} = \frac{\log_2 x}{1/4} = 4 \log_2 x \]
- Thus,
\[ 2^{\log_{2^{1/4}} x} = 2^{4 \log_2 x} = x^4 \]

2. **Second term:** \( 3^{\log_{27} (x^2 + 1)^3} \)
- Since \( 27 = 3^3 \):
\[ \log_{27} (x^2 + 1)^3 = \frac{3 \log_3 (x^2 + 1)}{3} = \log_3 (x^2 + 1) \]
- Thus,
\[ 3^{\log_{27} (x^2 + 1)^3} = (x^2 + 1) \]

3. **Combining the numerator:**
\[ \text{Numerator} = x^4 - (x^2 + 1) - 2x = x^4 - x^2 - 2x - 1 \]

**Denominator:**
1. **Denominator:** \( 7^{4 \log_{49} x} \)
- Since \( 49 = 7^2 \):
\[ \log_{49} x = \frac{\log_7 x}{\log_7 (7^2)} = \frac{\log_7 x}{2} \]
- Thus,
\[ 7^{4 \log_{49} x} = 7^{4 \cdot \frac{\log_7 x}{2}} = 7^{2 \log_7 x} = x^2 \]

2. **Combining the denominator:**
\[ \text{Denominator} = x^2 - x - 1 \]

### Step 2: Final expression for \( y \)
Putting it all together:
\[ y = \frac{x^4 - x^2 - 2x - 1}{x^2 - x - 1} \]

### Step 3: Differentiating \( y \)
Using the quotient rule:
\[ \frac{dy}{dx} = \frac{(u'v - uv')}{v^2} \]
where \( u = x^4 - x^2 - 2x - 1 \) and \( v = x^2 - x - 1 \).

1. **Finding \( u' \):**
\[ u' = 4x^3 - 2x - 2 \]

2. **Finding \( v' \):**
\[ v' = 2x - 1 \]

3. **Applying the quotient rule:**
\[ \frac{dy}{dx} = \frac{(4x^3 - 2x - 2)(x^2 - x - 1) - (x^4 - x^2 - 2x - 1)(2x - 1)}{(x^2 - x - 1)^2} \]

### Step 4: Simplifying \( \frac{dy}{dx} \)
After simplification, we find that:
\[ \frac{dy}{dx} = ax + b \]
where \( a = 2 \) and \( b = 1 \).

### Final Answer
The ordered pair \( (a, b) \) is:
\[ \boxed{(2, 1)} \]