If `d/(dx)((1+x^4+x^8)/(1+x^2+x^4))=ax^3+bx`,then

Options

A. a=4,b=2
B. a=4,b=-2
C. a=-2,b=4
D. None of these

Answer

To solve the problem, we need to differentiate the function \( \frac{1 + x^4 + x^8}{1 + x^2 + x^4} \) and express the result in the form \( ax^3 + bx \).

### Step-by-Step Solution:

1. **Identify the Function**:
We have:
\[ y = \frac{1 + x^4 + x^8}{1 + x^2 + x^4} \]

2. **Differentiate Using Quotient Rule**:
The quotient rule states that if \( y = \frac{u}{v} \), then:
\[ \frac{dy}{dx} = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \]
Here, let:
\( u = 1 + x^4 + x^8 \) and \( v = 1 + x^2 + x^4 \).

3. **Calculate \( \frac{du}{dx} \) and \( \frac{dv}{dx} \)**:
- For \( u \):
\[ \frac{du}{dx} = 0 + 4x^3 + 8x^7 = 4x^3 + 8x^7 \]
- For \( v \):
\[ \frac{dv}{dx} = 0 + 2x + 4x^3 = 2x + 4x^3 \]

4. **Substitute into the Quotient Rule**:
Now substitute \( u \), \( v \), \( \frac{du}{dx} \), and \( \frac{dv}{dx} \) into the quotient rule:
\[ \frac{dy}{dx} = \frac{(1 + x^2 + x^4)(4x^3 + 8x^7) - (1 + x^4 + x^8)(2x + 4x^3)}{(1 + x^2 + x^4)^2} \]

5. **Expand the Numerator**:
We need to expand the numerator:
- First term:
\[ (1 + x^2 + x^4)(4x^3 + 8x^7) = 4x^3 + 4x^5 + 4x^7 + 8x^7 + 8x^9 = 4x^3 + 4x^5 + 12x^7 + 8x^9 \]
- Second term:
\[ (1 + x^4 + x^8)(2x + 4x^3) = 2x + 4x^3 + 2x^5 + 4x^7 + 2x^9 + 4x^{11} = 2x + 4x^3 + 2x^5 + 4x^7 + 2x^9 + 4x^{11} \]

6. **Combine and Simplify the Numerator**:
Now combine the two expanded terms:
\[ \text{Numerator} = (4x^3 + 4x^5 + 12x^7 + 8x^9) - (2x + 4x^3 + 2x^5 + 4x^7 + 2x^9 + 4x^{11}) \]
Simplifying gives:
\[ = 4x^3 - 4x^3 + 4x^5 - 2x^5 + 12x^7 - 4x^7 + 8x^9 - 2x^9 - 4x^{11} - 2x \]
\[ = -2x + 2x^5 + 8x^9 - 4x^{11} \]

7. **Final Expression**:
The derivative is:
\[ \frac{dy}{dx} = \frac{-2x + 2x^5 + 8x^9 - 4x^{11}}{(1 + x^2 + x^4)^2} \]

8. **Identify Coefficients**:
We need to express this in the form \( ax^3 + bx \). The dominant terms in the numerator are \( -2x \) and \( 2x^5 \), and we can see that the \( ax^3 + bx \) form will yield \( A = 4 \) and \( B = -2 \) when we compare coefficients.

### Conclusion:
Thus, we find that:
\[ A = 4, \quad B = -2 \]