Question

If \[ \frac{d}{dx} \left(\frac{1 + x^2 + x^4}{1 + x + x^2}\right) = ax + b, \] then \( (a,b) \) is:

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

Hint:

For differentiating fractions, use the quotient rule: \( \frac{f'g - fg'}{g^2} \).

Answer 1

The Correct Option is C

Step 1: Differentiating using quotient rule Using the quotient rule: \[ \frac{d}{dx} \left(\frac{f(x)}{g(x)}\right) = \frac{f'g - fg'}{g^2}. \] Let \( f(x) = 1 + x^2 + x^4 \), \( g(x) = 1 + x + x^2 \). Computing derivatives: \[ f'(x) = 2x + 4x^3, \quad g'(x) = 1 + 2x. \] Applying the quotient rule and simplifying: \[ \frac{(2x + 4x^3)(1 + x + x^2) - (1 + x^2 + x^4)(1 + 2x)}{(1 + x + x^2)^2}. \] Simplifying, we get: \[ 2x - 1. \] Thus, \( a = 2 \), \( b = -1 \).