Derivative of \( \displaystyle 3 \sqrt{x^{2} + 1} \)
Problem 2.146 · medium
Differentiate \( \displaystyle f(x) = 3 \sqrt{x^{2} + 1} \).
- \[ \frac{d}{d x} 3 \sqrt{x^{2} + 1} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = 3 \frac{d}{d x} \sqrt{x^{2} + 1} \]constant-multiple rewritePull the constant out of the derivative. Rewrite the square root as a fractional power.✓ Proved
- \[ = \frac{3 \frac{d}{d x} \left(x^{2} + 1\right)}{2 \sqrt{x^{2} + 1}} \]chainApply the chain rule.✓ Proved
- \[ = \frac{3 x}{\sqrt{x^{2} + 1}} \]derivative algebra simplifyDifferentiate the inner function. Simplify the coefficients (3 * 1/2 * 2 = 3). Rewrite the negative exponent as a fraction.✓ Proved
Answer \( \frac{3 x}{\sqrt{x^{2} + 1}} \)
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 x**2 + 1 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x**2 + 1 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x**2 + 1 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x**2 + 1 = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where 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 (error) — Step 4 applies both the power rule and the chain rule in a single line, violating the rule that each step must change only one thing. The label "chain" is also incomplete because the power rule is also being used.deepseek-r1:70b: passqwen3.6:27b-mlx: pass
Every verdict on record (15)
qwen3.6:27b-mlx: pass 2026-09-20gpt-oss:20b: fail (error) 2026-09-20 — Step 4 applies both the power rule and the chain rule in a single line, violating the rule that each step must change only one thing. The label "chain" is also incomplete because the power rule is also being used.qwen3.6:27b-mlx: pass 2026-09-20gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-19 — The solution correctly applies the constant multiple rule, rewrite rule, chain rule, and algebraic simplification in distinct steps. All labels are appropriate and the logic is sound.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: pass 2026-09-19deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: pass 2026-09-18deepseek-r1:70b: pass 2026-09-18gpt-oss:20b: pass 2026-09-18gpt-oss:20b: pass 2026-09-17deepseek-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.