Derivative of \( \displaystyle \frac{\sqrt{\left(2 x + 1\right)^{2} + 1}}{2} \)
Problem 2.899 · medium Video
Differentiate \( \displaystyle f(x) = \frac{\sqrt{\left(2 x + 1\right)^{2} + 1}}{2} \).
- \[ \frac{d}{d x} \frac{\sqrt{\left(2 x + 1\right)^{2} + 1}}{2} \]Start with the derivative of the function.✓ Proved
- \[ = \frac{\frac{d}{d x} \sqrt{\left(2 x + 1\right)^{2} + 1}}{2} \]constant-multiple rewritePull out the constant factor 1/2. Rewrite the square root as a power.✓ Proved
- \[ = \frac{\frac{d}{d x} \left(\left(2 x + 1\right)^{2} + 1\right)}{4 \sqrt{\left(2 x + 1\right)^{2} + 1}} \]chainApply the chain rule.✓ Proved
- \[ = \frac{\frac{d}{d x} \left(2 x + 1\right)^{2}}{4 \sqrt{\left(2 x + 1\right)^{2} + 1}} \]sumDifferentiate the sum inside the parenthesis.✓ Proved
- \[ = \frac{\left(4 x + 2\right) \frac{d}{d x} \left(2 x + 1\right)}{4 \sqrt{\left(2 x + 1\right)^{2} + 1}} \]powerApply the power rule to the squared term.✓ Proved
- \[ = \frac{8 x + 4}{4 \sqrt{\left(2 x + 1\right)^{2} + 1}} \]derivative algebraDifferentiate the linear term. Multiply the constants.✓ Proved
- \[ = \frac{2 x + 1}{\sqrt{\left(2 x + 1\right)^{2} + 1}} \]simplifySimplify the final expression.✓ Proved
Answer \( \frac{2 x + 1}{\sqrt{\left(2 x + 1\right)^{2} + 1}} \)
✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. How we verify
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | line 1 is the problem as stated |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (2*x + 1)**2 + 1 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (2*x + 1)**2 + 1 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (2*x + 1)**2 + 1 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (2*x + 1)**2 + 1 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (2*x + 1)**2 + 1 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (2*x + 1)**2 + 1 = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where (2*x + 1)**2 + 1 = 0 |
| answer, a second way | ✓ Proved | sympy 1.14.0 | SymPy differentiated f directly and got the stated answer |
Reviewers
gpt-oss:20b: fail (error) — Step 5 applies the sum rule but drops the derivative of the constant 1. The correct derivative of ((2*x + 1)**2 + 1) is 2*(2*x + 1)*2 + 0, so the constant term must be retained (though it vanishes).qwen3.6:27b-mlx: pass
Senior review claude-sonnet-5-5, 2026-10-13: pass — Every line was proved equivalent by SymPy and the rule labels are legitimate. Step 5 correctly applies the sum rule, with the constant's zero derivative absorbed. Line 8 turns 2*(2x+1)*2 into 4*(2x+1), which is a fair algebra step.
gpt-oss:20b: dismiss — Dropping the derivative of the constant 1 is part of the sum-rule step, since d/dx(1)=0 is trivially absorbed. Line 5 is equal to line 4 (SymPy proved it), and the objection also misstates the step, because the 2*(2x+1)*2 expansion happens in later lines.
Every verdict on record (7)
claude-sonnet-5-5: pass 2026-10-13 — Every line was proved equivalent by SymPy and the rule labels are legitimate. Step 5 correctly applies the sum rule, with the constant's zero derivative absorbed. Line 8 turns 2*(2x+1)*2 into 4*(2x+1), which is a fair algebra step.qwen3.6:27b-mlx: pass 2026-09-27gpt-oss:20b: fail (error) 2026-09-27 — Step 5 applies the sum rule but drops the derivative of the constant 1. The correct derivative of ((2*x + 1)**2 + 1) is 2*(2*x + 1)*2 + 0, so the constant term must be retained (though it vanishes).qwen3.6:27b-mlx: pass 2026-09-26gpt-oss:20b: fail (error) 2026-09-26 — Step 5 drops the derivative of the constant term +1. The sum rule should produce Derivative((2*x+1)**2, x)+Derivative(1, x), but the latter is zero and was omitted, violating the one‑rule‑per‑step rule.qwen3.6:27b-mlx: pass 2026-09-26gpt-oss:20b: pass 2026-09-26
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
gemma4:26b, checked 2026-09-27 with SymPy 1.14.0.