Derivative of \( \displaystyle \operatorname{atan}{\left(4 x^{2} \right)} \)
Problem 2.607 · hard
Differentiate \( \displaystyle f(x) = \operatorname{atan}{\left(4 x^{2} \right)} \).
- \[ \frac{d}{d x} \operatorname{atan}{\left(4 x^{2} \right)} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{\frac{d}{d x} 4 x^{2}}{16 x^{4} + 1} \]chainApply the chain rule for the arctangent function.✓ Proved
- \[ = \frac{8 x}{16 x^{4} + 1} \]product algebraDifferentiate the inner function 4*x**2. Simplify the denominator by squaring the term.✓ Proved
Answer \( \frac{8 x}{16 x^{4} + 1} \)
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 undefined where 16*x**4 + 1 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 16*x**4 + 1 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 16*x**4 + 1 = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where 16*x**4 + 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 3 applies the power rule (derivative of 4*x**2) but is labeled "product". The correct label should be "power" or "constant-multiple".qwen3.6:27b-mlx: fail (style) — Step 3 is labeled 'product', but the operation performed is differentiating the inner function (a power rule application), not applying the product rule. The label should be 'derivative' or 'power'.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (style) 2026-09-21 — Step 3 is labeled 'product', but the operation performed is differentiating the inner function (a power rule application), not applying the product rule. The label should be 'derivative' or 'power'.gpt-oss:20b: fail (style) 2026-09-21 — Step 3 applies the power rule (derivative of 4*x**2) but is labeled "product". The correct label should be "power" or "constant-multiple".qwen3.6:27b-mlx: fail (error) 2026-09-21 — Step 3 is labeled 'product', but the step actually computes the derivative of the inner function (4*x**2), which is a 'derivative' or 'power' rule application, not a product rule. The product rule is not applicable here as there is no product of functions being differentiated.gpt-oss:20b: fail (error) 2026-09-21 — Step 3 applies two rules at once: it replaces Derivative(4*x**2,x) with 8*x (derivative rule) and multiplies by the existing factor 1/(1+(4*x**2)**2). The label "product" is also inappropriate; the step is a simple substitution, not a product rule application.
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.