Derivative of \( \displaystyle \operatorname{atan}^{2}{\left(4 x + 2 \right)} \)
Problem 2.334 · medium
Differentiate \( \displaystyle f(x) = \operatorname{atan}^{2}{\left(4 x + 2 \right)} \).
- \[ \frac{d}{d x} \operatorname{atan}^{2}{\left(4 x + 2 \right)} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = 2 \operatorname{atan}{\left(4 x + 2 \right)} \frac{d}{d x} \operatorname{atan}{\left(4 x + 2 \right)} \]powerApply the power rule.✓ Proved
- \[ = \frac{2 \operatorname{atan}{\left(4 x + 2 \right)} \frac{d}{d x} \left(4 x + 2\right)}{\left(4 x + 2\right)^{2} + 1} \]chainApply the chain rule to the inner function.✓ Proved
- \[ = \frac{8 \operatorname{atan}{\left(4 x + 2 \right)}}{\left(4 x + 2\right)^{2} + 1} \]derivative algebraDifferentiate the linear term 4*x + 2. Simplify the expression by multiplying the constants.✓ Proved
- \[ = \frac{8 \operatorname{atan}{\left(4 x + 2 \right)}}{16 x^{2} + 16 x + 5} \]algebra simplifyExpand the denominator. Combine like terms in the denominator.✓ Proved
Answer \( \frac{8 \operatorname{atan}{\left(4 x + 2 \right)}}{\left(4 x + 2\right)^{2} + 1} \)
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 (4*x + 2)**2 + 1 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (4*x + 2)**2 + 1 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (4*x + 2)**2 + 1 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (4*x + 2)**2 + 1 = 0 undefined where 16*x**2 + 16*x + 5 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 16*x**2 + 16*x + 5 = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where (4*x + 2)**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: passdeepseek-r1:70b: passqwen3.6:27b-mlx: pass
Every verdict on record (13)
qwen3.6:27b-mlx: pass 2026-09-20gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-20gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-19deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: fail (error) 2026-09-19 — Step 6 expands the denominator but omits the +1 term, effectively applying two rules (expansion and addition) in one step and producing an incorrect denominator. The correct expansion is 1 + (4*x + 2)**2 = 16*x**2 + 16*x + 5.qwen3.6:27b-mlx: pass 2026-09-19deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: pass 2026-09-18deepseek-r1:70b: pass 2026-09-18gpt-oss:20b: pass 2026-09-18
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.