Derivative of \( \displaystyle \frac{\sqrt{9 x^{2} + 1}}{3} \)
Problem 2.57 · medium
Differentiate \( \displaystyle f(x) = \frac{\sqrt{9 x^{2} + 1}}{3} \).
- \[ \frac{d}{d x} \frac{\sqrt{9 x^{2} + 1}}{3} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{\frac{d}{d x} \sqrt{9 x^{2} + 1}}{3} \]constant rewritePull out the constant factor 1/3. Rewrite the square root as a fractional power.✓ Proved
- \[ = \frac{\frac{d}{d x} \left(9 x^{2} + 1\right)}{6 \sqrt{9 x^{2} + 1}} \]chain algebraApply the chain rule. Simplify the constant coefficient.✓ Proved
- \[ = \frac{\frac{d}{d x} 1 + \frac{d}{d x} 9 x^{2}}{6 \sqrt{9 x^{2} + 1}} \]sumApply the sum rule to the inner function.✓ Proved
- \[ = \frac{3 x}{\sqrt{9 x^{2} + 1}} \]derivative algebra simplify rewriteDifferentiate each term in the sum. Simplify the expression. Simplify the final expression. Rewrite with a positive exponent.✓ Proved
Answer \( \frac{3 x}{\sqrt{9 x^{2} + 1}} \)
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**2 + 1 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 9*x**2 + 1 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 9*x**2 + 1 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 9*x**2 + 1 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 9*x**2 + 1 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 9*x**2 + 1 = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 9*x**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**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 (style) — Step 5 applies two rules (algebraic simplification of the constant and a rewrite of 9*x**2 as 9*x*x) but is labeled only as "algebra", violating the one‑rule‑per‑step rule.deepseek-r1:70b: fail (misleading) — Step 5 mislabels the application of the sum rule as 'algebra'.qwen3.6:27b-mlx: pass
Every verdict on record (12)
qwen3.6:27b-mlx: pass 2026-09-20gpt-oss:20b: fail (style) 2026-09-20 — Step 5 applies two rules (algebraic simplification of the constant and a rewrite of 9*x**2 as 9*x*x) but is labeled only as "algebra", violating the one‑rule‑per‑step rule.qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies the constant multiple rule, chain rule, and power rule in separate steps. All labels are valid and the logic is sound.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-19 — The solution correctly applies the constant multiple rule, chain rule, and power rule in a logical sequence. Each step modifies only one aspect of the expression and uses valid labels from the fixed vocabulary.deepseek-r1:70b: fail (misleading) 2026-09-19 — Step 5 mislabels the application of the sum rule as 'algebra'.gpt-oss:20b: fail (style) 2026-09-19 — Step 5 applies two transformations at once: it both simplifies the constant coefficient (1/3·1/2→1/6) and rewrites the inner expression 9*x**2 as 9*x*x. Each step must change only one thing, so this step violates the granularity rule.qwen3.6:27b-mlx: pass 2026-09-19 — The solution correctly applies differentiation rules and algebraic simplifications in a logical sequence. Each step adheres to the single-change constraint and uses valid labels from the provided vocabulary.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: fail (error) 2026-09-19 — Step 5 to 6 applies two rules at once: it rewrites 9*x**2 as 9*x*x (rewrite) and then applies the sum rule to split the inner function into 9*x*x + 1. Each step must change only one thing.gpt-oss:20b: fail 2026-09-17 — Step 5 applies two operations—simplifying the constant coefficient and rewriting the inner function (9*x**2 to 9*x*x)—but only the constant simplification is noted, mislabeling the step.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.