Derivative of \( \displaystyle \frac{3 \operatorname{atan}{\left(4 x \right)}}{4} \)
Problem 2.694 · medium
Differentiate \( \displaystyle f(x) = \frac{3 \operatorname{atan}{\left(4 x \right)}}{4} \).
- \[ \frac{d}{d x} \frac{3 \operatorname{atan}{\left(4 x \right)}}{4} \]Start with the derivative of the function.✓ Proved
- \[ = \frac{3 \frac{d}{d x} \operatorname{atan}{\left(4 x \right)}}{4} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = \frac{3 \frac{d}{d x} 4 x}{4 \left(16 x^{2} + 1\right)} \]chainApply the chain rule to the arctangent function.✓ Proved
- \[ = \frac{3}{16 x^{2} + 1} \]power algebraSquare the term inside the denominator. Simplify the expression by canceling the 4s.✓ Proved
Answer \( \frac{3}{16 x^{2} + 1} \)
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 |
| 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 |
| 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 two separate changes—simplifying the squared term and evaluating the derivative of 4*x—yet it is labeled only as "power". Each step must modify a single rule, and the label does not match the operations performed.qwen3.6:27b-mlx: fail (style) — Step 4 is labeled 'power' but performs the differentiation of 4*x (which is a 'derivative' or 'constant-multiple' step), not just squaring. The note 'Square the term inside the denominator' is also misleading because the squaring happened in the previous step's formula application, while this step computes the inner derivative.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (style) 2026-09-21 — Step 4 is labeled 'power' but performs the differentiation of 4*x (which is a 'derivative' or 'constant-multiple' step), not just squaring. The note 'Square the term inside the denominator' is also misleading because the squaring happened in the previous step's formula application, while this step computes the inner derivative.gpt-oss:20b: fail (error) 2026-09-21 — Step 4 applies two separate changes—simplifying the squared term and evaluating the derivative of 4*x—yet it is labeled only as "power". Each step must modify a single rule, and the label does not match the operations performed.qwen3.6:27b-mlx: fail (error) 2026-09-21 — Step 4 is labeled 'power' but performs the differentiation of 4*x (which is a 'derivative' or 'constant-multiple' step), not just squaring the term. The note 'Square the term inside the denominator' describes an algebraic simplification that was already present in Step 3, ignoring the actual operation performed (differentiation).gpt-oss:20b: fail (error) 2026-09-21 — Step 4 applies two rules at once (simplifying (4*x)**2 to 16*x**2 and multiplying by 4) and is labeled incorrectly as "power"; it should be split into separate algebraic simplification and multiplication steps with appropriate labels.
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.