Derivative of \( \displaystyle - \frac{\sqrt{\left(4 x + 1\right)^{2} + 1}}{4} \)
Problem 2.163 · medium
Differentiate \( \displaystyle f(x) = - \frac{\sqrt{\left(4 x + 1\right)^{2} + 1}}{4} \).
- \[ \frac{d}{d x} \left(- \frac{\sqrt{\left(4 x + 1\right)^{2} + 1}}{4}\right) \]Start with the derivative of the function.✓ Proved
- \[ = - \frac{\frac{d}{d x} \sqrt{\left(4 x + 1\right)^{2} + 1}}{4} \]constant rewritePull out the constant factor. Rewrite the square root as a fractional power.✓ Proved
- \[ = - \frac{\frac{d}{d x} \left(\left(4 x + 1\right)^{2} + 1\right)}{8 \sqrt{\left(4 x + 1\right)^{2} + 1}} \]chainApply the chain rule.✓ Proved
- \[ = - \frac{\frac{d}{d x} \left(4 x + 1\right)^{2}}{8 \sqrt{\left(4 x + 1\right)^{2} + 1}} \]algebraSimplify the constant coefficients.✓ Proved
- \[ = - \frac{\left(8 x + 2\right) \frac{d}{d x} \left(4 x + 1\right)}{8 \sqrt{\left(4 x + 1\right)^{2} + 1}} \]powerApply the power rule and chain rule to the inner squared term.✓ Proved
- \[ = - \frac{32 x + 8}{8 \sqrt{\left(4 x + 1\right)^{2} + 1}} \]derivative algebraDifferentiate the innermost linear term. Multiply the constants in the second part.✓ Proved
- \[ = \frac{- 4 x - 1}{\sqrt{\left(4 x + 1\right)^{2} + 1}} \]algebra simplifySimplify the expression by combining constants and moving the negative exponent. Final simplified form.✓ Proved
Answer \( \frac{- 4 x - 1}{\sqrt{\left(4 x + 1\right)^{2} + 1}} \)
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. The reviewers disagree about how one step is explained; every verdict is in 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 |
| 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: passdeepseek-r1:70b: passqwen3.6:27b-mlx: fail (style) — Step 6 is labeled 'power' but the note explicitly states 'Apply the power rule and chain rule', violating the single-rule-per-step constraint. Step 5 is labeled 'algebra' but performs differentiation of the constant term inside the derivative operator (Derivative(... + 1, x) -> Derivative(..., x)), which is a differentiation step, not just algebra.
Every verdict on record (15)
qwen3.6:27b-mlx: fail (style) 2026-09-20 — Step 6 is labeled 'power' but the note explicitly states 'Apply the power rule and chain rule', violating the single-rule-per-step constraint. Step 5 is labeled 'algebra' but performs differentiation of the constant term inside the derivative operator (Derivative(... + 1, x) -> Derivative(..., x)), which is a differentiation step, not just algebra.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: fail (style) 2026-09-20 — Step 6 is labeled 'power' but the note explicitly states 'Apply the power rule and chain rule', violating the single-rule-per-step constraint. Step 5 is labeled 'algebra' but performs differentiation (dropping the derivative of the constant 1), which should be labeled 'derivative' or 'sum'.gpt-oss:20b: fail (error) 2026-09-20 — Step 5 incorrectly drops the derivative of the constant term 1. The derivative of ((4*x+1)**2 + 1) should be handled by the sum rule, giving Derivative((4*x+1)**2, x) + 0, not simply Derivative((4*x+1)**2, x).qwen3.6:27b-mlx: fail (style) 2026-09-19 — Step 6 applies both the power rule and the chain rule simultaneously, violating the one-rule-per-step constraint. Step 5 is labeled 'algebra' but performs differentiation (dropping the derivative of the constant 1), which should be labeled 'derivative' or split.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: fail (error) 2026-09-19 — The derivative of the constant term +1 was omitted. Step 4 correctly has Derivative((4*x + 1)**2 + 1, x), but step 5 replaces it with Derivative((4*x + 1)**2, x), dropping the derivative of the constant. This is a mathematical error that propagates through the rest of the solution.qwen3.6:27b-mlx: fail (style) 2026-09-19 — Step 6 applies both the power rule and the chain rule simultaneously, violating the single-rule-per-step constraint. It should be split into a power rule step followed by a chain rule step.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: fail (error) 2026-09-19 — Step 5 combines two simplifications (removing the derivative of the constant and simplifying the derivative of a sum) in one line, which violates the one‑rule‑per‑step rule. Step 6 applies both the power rule and the chain rule to the inner squared term in a single step, also a multi‑rule application.qwen3.6:27b-mlx: fail (style) 2026-09-18 — Step 6 applies both the power rule and the chain rule simultaneously, violating the single-rule-per-step constraint. Step 5 is also defective as it combines constant simplification with the removal of the derivative of the constant term inside the sum, which should be handled in separate steps.deepseek-r1:70b: fail (style) 2026-09-18 — Step 1 is missing a rule label.gpt-oss:20b: fail (error) 2026-09-18 — Step 5 incorrectly drops the derivative of the constant term +1; the derivative of ((4*x + 1)**2 + 1) should be 2*(4*x + 1)*4 + 0, not just 2*(4*x + 1)*4.gpt-oss:20b: fail 2026-09-17 — Step 5 combines two operations—simplifying the constant factor and dropping the derivative of the constant term—yet labels it only as “algebra.” This mislabels the rule applied and obscures the use of the derivative of a constant, which is a separate rule.deepseek-r1:70b: pass 2026-09-17
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.