Derivative of \( \displaystyle - \frac{\sqrt{\left(4 x - 1\right)^{2} + 1}}{4} \)
Problem 2.563 · 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-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 \( \frac{1 - 4 x}{\sqrt{\left(4 x - 1\right)^{2} + 1}} \)
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 |
| 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 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.
Every verdict on record (4)
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.