Derivative of \( \displaystyle - \frac{\sqrt{\left(4 x - 1\right)^{2} + 1}}{4} \)
Problem 2.563 · medium Video
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-multiple rewritePull out the constant factor. Rewrite the square root as a fractional exponent.✓ 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
- \[ = - \frac{\frac{d}{d x} \left(4 x - 1\right)^{2}}{8 \sqrt{\left(4 x - 1\right)^{2} + 1}} \]sumDifferentiate the sum inside the parentheses.✓ 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 squared term.✓ Proved
- \[ = - \frac{32 x - 8}{8 \sqrt{\left(4 x - 1\right)^{2} + 1}} \]derivative algebraDifferentiate the linear term. Multiply the constants.✓ Proved
- \[ = \frac{1 - 4 x}{\sqrt{\left(4 x - 1\right)^{2} + 1}} \]simplifySimplify the final expression.✓ Proved
Answer \( \frac{1 - 4 x}{\sqrt{\left(4 x - 1\right)^{2} + 1}} \)
✓ 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 (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 |
| 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 applies the sum rule but omits the derivative of the constant term 1, resulting in an incomplete expression. The derivative of 1 is 0, so the missing term does not affect the final result, but the step still violates the rule‑by‑one‑change requirement.qwen3.6:27b-mlx: fail (error) — Step 6 is labeled 'power' but applies both the power rule and the chain rule simultaneously, violating the one-rule-per-step constraint. Additionally, Step 5 is labeled 'sum' but performs the differentiation of the inner function (removing the derivative operator), which is a differentiation step, not a presentational sum rule application.
Senior review claude-sonnet-5-5, 2026-10-10: pass — All lines are SymPy-proved equal, the final line matches the stated answer, and the rule labels each name a rule actually applied. The panel objections are about granularity and are not real defects.
gpt-oss:20b: dismiss — Step 5 legitimately applies the sum rule, and dropping the derivative of the constant 1 (which is 0) is part of the same step; SymPy proved the lines equal, so nothing false is stated.qwen3.6:27b-mlx: dismiss — Applying the power rule together with the evaluation of the simple inner derivative is treated as one chain-style step under the contract, 'power' names a rule that was used, and the 'sum' label in step 5 is valid because the sum rule was applied and the constant's derivative vanishes.
Every verdict on record (7)
claude-sonnet-5-5: pass 2026-10-10 — All lines are SymPy-proved equal, the final line matches the stated answer, and the rule labels each name a rule actually applied. The panel objections are about granularity and are not real defects.qwen3.6:27b-mlx: fail (error) 2026-09-26 — Step 6 is labeled 'power' but applies both the power rule and the chain rule simultaneously, violating the one-rule-per-step constraint. Additionally, Step 5 is labeled 'sum' but performs the differentiation of the inner function (removing the derivative operator), which is a differentiation step, not a presentational sum rule application.gpt-oss:20b: fail (error) 2026-09-26 — Step 5 applies the sum rule but omits the derivative of the constant term 1, resulting in an incomplete expression. The derivative of 1 is 0, so the missing term does not affect the final result, but the step still violates the rule‑by‑one‑change requirement.qwen3.6:27b-mlx: fail (error) 2026-09-21 — Step 5 is labeled 'sum' but performs differentiation of the inner term, which is not a 'sum' rule application in the context of differentiation rules (it's part of the chain rule expansion or linearity). More critically, Step 6 is labeled 'power' but the note explicitly states 'Apply the power rule and chain rule', and the step itself applies both the power rule to the outer square and the chain rule to the inner linear term. Applying two rules (power and chain) in a single step violates the 'one rule per step' constraint.gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: fail (style) 2026-09-21 — Step 5 is labeled 'sum' but performs the differentiation of the inner function (specifically, dropping the constant term inside the derivative), which is a differentiation step, not a presentational sum rule. Step 6 is labeled 'power' but applies both the power rule and the chain rule simultaneously, violating the one-rule-per-step constraint.gpt-oss:20b: fail (error) 2026-09-21 — Step 5 applies the sum rule but omits the derivative of the constant term 1, which is zero. The correct derivative inside the parentheses should be 2*(4*x-1)*4 + 0, but the zero term is missing, making the step mathematically incomplete.
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.