Derivative of \( \displaystyle \frac{3 \sqrt{4 \left(x + 1\right)^{2} + 1}}{2} \)
Problem 2.862 · medium
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.
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: fail (error) — 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) — 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.
Every verdict on record (4)
qwen3.6:27b-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 2026-09-26 with SymPy 1.14.0.