Derivative of \( \displaystyle - \frac{\sqrt{9 \left(x - 1\right)^{2} + 1}}{3} \)

Problem 2.7 · medium Video

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 \( \frac{3 \left(1 - x\right)}{\sqrt{9 \left(x - 1\right)^{2} + 1}} \)

Top: the graph of f = -sqrt(9(x - 1)² + 1)/3. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = -(9x - 9)/(3sqrt(9(x - 1)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 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: pass
gpt-oss:20b: pass
qwen3.6:27b-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.

Senior review claude-sonnet-5-5, 2026-10-03: pass — The derivative -3(x-1)/sqrt(9(x-1)^2+1) is correct and equals the stated answer 3(1-x)/sqrt(...). Every line was proved by SymPy and the rule labels are legitimate.

qwen3.6:27b-mlx: dismiss — Step 9 differentiates 9*(x-1)**2 as a single change, and the power rule applied to (x-1)**2 with a leaving inner derivative is one legitimate step; the label 'power' names a rule that was applied, and 'chain' would also have been acceptable.

Every verdict on record (12)

claude-sonnet-5-5: pass 2026-10-03 — The derivative -3(x-1)/sqrt(9(x-1)^2+1) is correct and equals the stated answer 3(1-x)/sqrt(...). Every line was proved by SymPy and the rule labels are legitimate.
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 2026-09-20
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 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 2026-09-20
qwen3.6:27b-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 2026-09-19
gpt-oss:20b: pass 2026-09-19
qwen3.6:27b-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 2026-09-19
gpt-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.