Derivative of \( \displaystyle \frac{3 \operatorname{atan}{\left(4 x \right)}}{4} \)
Problem 2.694 · medium Video
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 \( \frac{3}{16 x^{2} + 1} \)
✓ 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 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 (style) — Step 4 incorrectly labels the simplification of (4*x)**2 to 16*x**2 as a "power" rule; it is an algebraic simplification. Similarly, step 5’s cancellation of the 4’s is an algebraic simplification, not a constant‑multiple rule. These mislabelings violate the one‑rule‑per‑step requirement.qwen3.6:27b-mlx: fail (error) — Step 4 is labeled 'power' but performs the differentiation of the inner function 4*x (which requires the 'derivative' rule), not just squaring the term. The label does not match the operation performed.
Senior review claude-sonnet-5-5, 2026-10-11: fail — The math is correct and the answer matches. Step 4 both squares (4*x) and evaluates the inner derivative 4*x -> 4 under a single 'power' label. These should be separate steps, or the derivative evaluation should be done in step 3.
gpt-oss:20b: dismiss — Labelling the expansion of (4*x)**2 as 'power' is defensible, and the claim that step 5 is labelled constant-multiple is false, since it is labelled algebra.qwen3.6:27b-mlx: uphold — Step 4 also evaluates Derivative(4*x, x) to 4, which step 3 left unevaluated; 'power' and the note 'square the term' do not cover that, so the step does two independent things.
Every verdict on record (7)
claude-sonnet-5-5: fail (style) 2026-10-11 — The math is correct and the answer matches. Step 4 both squares (4*x) and evaluates the inner derivative 4*x -> 4 under a single 'power' label. These should be separate steps, or the derivative evaluation should be done in step 3.qwen3.6:27b-mlx: fail (error) 2026-09-26 — Step 4 is labeled 'power' but performs the differentiation of the inner function 4*x (which requires the 'derivative' rule), not just squaring the term. The label does not match the operation performed.gpt-oss:20b: fail (style) 2026-09-26 — Step 4 incorrectly labels the simplification of (4*x)**2 to 16*x**2 as a "power" rule; it is an algebraic simplification. Similarly, step 5’s cancellation of the 4’s is an algebraic simplification, not a constant‑multiple rule. These mislabelings violate the one‑rule‑per‑step requirement.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.