Derivative of \( \displaystyle \frac{3 \sqrt{4 \left(x + 1\right)^{2} + 1}}{2} \)
Problem 2.862 · medium Video
Differentiate \( \displaystyle f(x) = \frac{3 \sqrt{4 \left(x + 1\right)^{2} + 1}}{2} \).
- \[ \frac{d}{d x} \frac{3 \sqrt{4 \left(x + 1\right)^{2} + 1}}{2} \]Differentiate the function with respect to x.✓ Proved
- \[ = \frac{3 \frac{d}{d x} \sqrt{4 \left(x + 1\right)^{2} + 1}}{2} \]constant-multiple rewritePull out the constant factor. Rewrite the square root as a power.✓ Proved
- \[ = \frac{3 \frac{d}{d x} \left(4 \left(x + 1\right)^{2} + 1\right)}{4 \sqrt{4 \left(x + 1\right)^{2} + 1}} \]chainApply the chain rule.✓ Proved
- \[ = \frac{3 \frac{d}{d x} 4 \left(x + 1\right)^{2}}{4 \sqrt{4 \left(x + 1\right)^{2} + 1}} \]sum constantDifferentiate the sum inside the parenthesis. The derivative of the constant 1 is 0.✓ Proved
- \[ = \frac{3 \frac{d}{d x} \left(x + 1\right)^{2}}{\sqrt{4 \left(x + 1\right)^{2} + 1}} \]constant-multiple simplifyPull out the constant factor 4. Simplify the coefficients (3/4 * 4 = 3).✓ Proved
- \[ = \frac{6 \left(x + 1\right)}{\sqrt{4 \left(x + 1\right)^{2} + 1}} \]powerApply the power rule to (x + 1)**2.✓ Proved
- \[ = \frac{6 \left(x + 1\right) \frac{d}{d x} \left(x + 1\right)}{\sqrt{4 \left(x + 1\right)^{2} + 1}} \]constant-multiplePull out the constant factor 2.✓ Proved
- \[ = \frac{6 \left(x + 1\right)}{\sqrt{4 \left(x + 1\right)^{2} + 1}} \]derivativeThe derivative of x + 1 is 1.✓ Proved
- \[ = \frac{6 x + 6}{\sqrt{4 \left(x + 1\right)^{2} + 1}} \]simplifySimplify the final expression.✓ Proved
Answer \( \frac{6 \left(x + 1\right)}{\sqrt{4 \left(x + 1\right)^{2} + 1}} \)
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. A reviewer's objection to the explanation stands; see the receipt. 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 4*(x + 1)**2 + 1 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*(x + 1)**2 + 1 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*(x + 1)**2 + 1 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*(x + 1)**2 + 1 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*(x + 1)**2 + 1 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*(x + 1)**2 + 1 = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*(x + 1)**2 + 1 = 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*(x + 1)**2 + 1 = 0 |
| 12 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*(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 4*(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: passqwen3.6:27b-mlx: fail (error) — Step 9 applies the chain rule to differentiate (x+1)^2 but labels it 'power'. The power rule alone yields 2(x+1), but the step produces 2(x+1)*1, implying the derivative of the inner function was taken. This is a chain rule application mislabeled as power rule. Additionally, Step 10 is redundant and confusingly labeled 'constant-multiple' when it appears to just re-insert the derivative operator for clarity, but the primary error is in Step 9.
Senior review claude-sonnet-5-5, 2026-10-13: fail — The math is correct at every line and the final answer matches. However, line 9 already evaluates the inner derivative to 1, and line 10 then undoes that by reintroducing Derivative(x+1, x) under the false label 'constant-multiple' (nothing is pulled out). Line 11 then re-evaluates it, so the steps are out of order and mislabelled.
qwen3.6:27b-mlx: uphold — The step 9 complaint is wrong, since power rule with the inner derivative 1 evaluated is acceptable under the chain-evaluation allowance. The step 10 complaint is correct: it only swaps the 1 back to Derivative(x+1, x) and is labelled 'constant-multiple' with a note about pulling out a factor 2 that was already outside, so the label and note are false.
Every verdict on record (7)
claude-sonnet-5-5: fail (misleading) 2026-10-13 — The math is correct at every line and the final answer matches. However, line 9 already evaluates the inner derivative to 1, and line 10 then undoes that by reintroducing Derivative(x+1, x) under the false label 'constant-multiple' (nothing is pulled out). Line 11 then re-evaluates it, so the steps are out of order and mislabelled.qwen3.6:27b-mlx: fail (error) 2026-09-27 — Step 9 applies the chain rule to differentiate (x+1)^2 but labels it 'power'. The power rule alone yields 2(x+1), but the step produces 2(x+1)*1, implying the derivative of the inner function was taken. This is a chain rule application mislabeled as power rule. Additionally, Step 10 is redundant and confusingly labeled 'constant-multiple' when it appears to just re-insert the derivative operator for clarity, but the primary error is in Step 9.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: fail (error) 2026-09-26 — Step 9 applies the power rule but fails to apply the chain rule for the inner function (x+1), omitting the derivative of the inner term in the result. Step 10 incorrectly labels the insertion of 'Derivative(x + 1, x)' as 'constant-multiple'; this is a structural change required by the chain rule, not a constant factor extraction.gpt-oss:20b: fail (error) 2026-09-26 — Step 5 incorrectly applies the sum rule by dropping the derivative of the constant 1, and step 6 replaces that constant with 0. The derivative of 4*(x+1)^2+1 should be 8*(x+1), not just the derivative of the quadratic term.qwen3.6:27b-mlx: fail (error) 2026-09-26 — Step 9 applies the chain rule to differentiate (x+1)^2 but labels it 'power', which is incorrect as the power rule alone does not account for the inner derivative. Step 10 incorrectly labels the separation of the inner derivative as 'constant-multiple'; it is actually the second part of the chain rule application.gpt-oss:20b: fail (error) 2026-09-26 — Step 9 applies both the power rule and the derivative of the inner function in one line, violating the one‑rule‑per‑step rule. Step 10 similarly combines a constant‑multiple extraction with a derivative, also a multi‑rule step.
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.