Derivative of \( \displaystyle \ln{\left(16 x^{2} + 1 \right)} \)
Problem 2.370 · hard
Differentiate \( \displaystyle f(x) = \ln{\left(16 x^{2} + 1 \right)} \).
- \[ \frac{d}{d x} \ln{\left(16 x^{2} + 1 \right)} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{\frac{d}{d x} \left(16 x^{2} + 1\right)}{16 x^{2} + 1} \]chainApply the chain rule for the natural logarithm.✓ Proved
- \[ = \frac{\frac{d}{d x} 1 + \frac{d}{d x} 16 x^{2}}{16 x^{2} + 1} \]sumDifferentiate the sum inside the parentheses.✓ Proved
- \[ = \frac{32 x}{16 x^{2} + 1} \]derivative constant algebraDifferentiate each term separately. The derivative of the constant 1 is 0. Multiply the terms together.✓ Proved
Answer \( \frac{32 x}{16 x^{2} + 1} \)
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 log is undefined for non-positive arguments undefined where 16*x**2 + 1 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 16*x**2 + 1 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 16*x**2 + 1 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 16*x**2 + 1 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 16*x**2 + 1 = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where 16*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 the derivative rule to two terms at once (32*x + 0), violating the rule‑by‑step requirement that each step change only one thing.deepseek-r1:70b: passqwen3.6:27b-mlx: pass
Every verdict on record (10)
qwen3.6:27b-mlx: pass 2026-09-20gpt-oss:20b: fail (error) 2026-09-20 — Step 4 applies the derivative rule to two terms at once (32*x + 0), violating the rule‑by‑step requirement that each step change only one thing.qwen3.6:27b-mlx: pass 2026-09-20gpt-oss:20b: fail (style) 2026-09-20 — Step 5 removes the zero term using simplification, but labels it as "constant". The correct label for eliminating the zero is "simplify".qwen3.6:27b-mlx: pass 2026-09-19deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: pass 2026-09-19 — The solution correctly applies the chain rule, sum rule, and power rule in distinct steps. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19
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.