If \frac{d}{dx}(\frac{1+x^{4}+x^{8}}{1-x^{2}+x^{4}})=ax+bx^{3}, then which one of the following is correct?
- A. a=b
- B. a=2b
- C. a+b=0
- D. 2a=b ✓
Correct Answer: D. 2a=b
Explanation
The numerator can be factored algebraically: 1+x^4+x^8 = (x^4+1)^2 - x^4 = (x^4+x^2+1)(x^4-x^2+1). Dividing by the denominator leaves x^4+x^2+1. Differentiating this with respect to x yields 4x^3+2x. Comparing this with ax+bx^3, we find a=2 and b=4. Therefore, 2a=b.
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