Derivative of \( \displaystyle - \frac{\sqrt{9 \left(x - 1\right)^{2} + 1}}{3} \)
Problem 2.7 · medium
Differentiate \( \displaystyle f(x) = - \frac{\sqrt{9 \left(x - 1\right)^{2} + 1}}{3} \).
- \[ \frac{d}{d x} \left(- \frac{\sqrt{9 \left(x - 1\right)^{2} + 1}}{3}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = - \frac{\frac{d}{d x} \sqrt{9 \left(x - 1\right)^{2} + 1}}{3} \]constant rewritePull out the constant factor. Rewrite the square root as a fractional power.✓ Proved
- \[ = - \frac{\frac{d}{d x} \left(9 \left(x - 1\right)^{2} + 1\right)}{6 \sqrt{9 \left(x - 1\right)^{2} + 1}} \]chain algebraApply the chain rule. Multiply the constants.✓ Proved
- \[ = - \frac{\frac{d}{d x} 1 + \frac{d}{d x} 9 \left(x - 1\right)^{2}}{6 \sqrt{9 \left(x - 1\right)^{2} + 1}} \]sumApply the sum rule.✓ Proved
- \[ = - \frac{\frac{d}{d x} 9 \left(x - 1\right)^{2}}{6 \sqrt{9 \left(x - 1\right)^{2} + 1}} \]derivative algebraThe derivative of a constant is zero. Simplify the expression.✓ Proved
- \[ = - \frac{\left(18 x - 18\right) \frac{d}{d x} \left(x - 1\right)}{6 \sqrt{9 \left(x - 1\right)^{2} + 1}} \]powerApply the power rule and chain rule to the inner term.✓ Proved
- \[ = - \frac{18 x - 18}{6 \sqrt{9 \left(x - 1\right)^{2} + 1}} \]derivativeThe derivative of (x - 1) is 1.✓ Proved
- \[ = \frac{3 - 3 x}{\sqrt{9 \left(x - 1\right)^{2} + 1}} \]algebra simplify rewriteMultiply the constants. Simplify the coefficient. Rewrite using a square root in the denominator.✓ Proved
Answer \( \frac{3 \left(1 - x\right)}{\sqrt{9 \left(x - 1\right)^{2} + 1}} \)
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. 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 9*(x - 1)**2 + 1 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 9*(x - 1)**2 + 1 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 9*(x - 1)**2 + 1 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 9*(x - 1)**2 + 1 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 9*(x - 1)**2 + 1 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 9*(x - 1)**2 + 1 = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 9*(x - 1)**2 + 1 = 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 9*(x - 1)**2 + 1 = 0 |
| 12 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 9*(x - 1)**2 + 1 = 0 |
| 13 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 9*(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 9*(x - 1)**2 + 1 = 0 |
| answer, a second way | ✓ Proved | sympy 1.14.0 | SymPy differentiated f directly and got the stated answer |
Reviewers
deepseek-r1:70b: passgpt-oss:20b: passqwen3.6:27b-mlx: fail (style) — Step 9 applies both the power rule and the chain rule simultaneously, violating the one-rule-per-step constraint. It should be split into a power rule step followed by a chain rule step.
Every verdict on record (11)
qwen3.6:27b-mlx: fail (style) 2026-09-20 — Step 9 applies both the power rule and the chain rule simultaneously, violating the one-rule-per-step constraint. It should be split into a power rule step followed by a chain rule step.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: fail (style) 2026-09-20 — Step 9 applies both the power rule and the chain rule simultaneously, violating the one-rule-per-step constraint. It should be split into a step applying the power rule to the outer function and a subsequent step applying the chain rule to the inner function.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: fail (style) 2026-09-19 — Step 9 applies both the power rule and the chain rule simultaneously, violating the constraint that each step must change only one thing. It should be split into a power rule step followed by a chain rule step.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: fail (error) 2026-09-19 — Step 9 applies both the power rule and the chain rule simultaneously, violating the constraint that each step must change only one thing. 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 (style) 2026-09-19 — Step 9 applies both the power rule and the constant‑multiple rule at once but is labeled only as "power". Each step should change only one rule, so the label or granularity should be corrected.deepseek-r1:70b: pass 2026-09-16
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.