Find the derivative of the rational function f(x) = (3x - 5) / (2x + 1) for x != -1/2.
Answer and explanation
Correct answer: B. 13 / (2x + 1)^2
- Step 1: Apply the quotient rule: f'(x) = [u'v - uv'] / v^2, where u = 3x - 5 and v = 2x + 1.
- Step 2: Compute derivatives: u' = 3 and v' = 2.
- Step 3: Calculate numerator: 3(2x + 1) - 2(3x - 5) = 6x + 3 - 6x + 10 = 13.
- Step 4: Combine terms: f'(x) = 13 / (2x + 1)^2.
Answer
13 / (2x + 1)^2.
Why not the others: '3 / (2x + 1)^2' differentiates only the numerator without completing the quotient rule; '(12x - 7) / (2x + 1)^2' adds products rather than subtracting; '-7 / (2x + 1)^2' miscalculates 3 - (-10) as 3 - 10 = -7.
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