Derivative of \( \displaystyle \left(2 x - 3\right) \operatorname{atan}{\left(2 x - 3 \right)} \)
Problem 2.650 · medium
Differentiate \( \displaystyle f(x) = \left(2 x - 3\right) \operatorname{atan}{\left(2 x - 3 \right)} \).
- \[ \frac{d}{d x} \left(2 x - 3\right) \operatorname{atan}{\left(2 x - 3 \right)} \]Start with the derivative of the function.✓ Proved
- \[ = \left(2 x - 3\right) \frac{d}{d x} \operatorname{atan}{\left(2 x - 3 \right)} + \operatorname{atan}{\left(2 x - 3 \right)} \frac{d}{d x} \left(2 x - 3\right) \]productApply the product rule.✓ Proved
- \[ = \left(2 x - 3\right) \frac{d}{d x} \operatorname{atan}{\left(2 x - 3 \right)} + 2 \operatorname{atan}{\left(2 x - 3 \right)} \]derivativeDifferentiate the second part of the product.✓ Proved
- \[ = \frac{2 \left(2 x - 3\right)}{\left(2 x - 3\right)^{2} + 1} + 2 \operatorname{atan}{\left(2 x - 3 \right)} \]inverse-trigApply the derivative rule for arctan.✓ Proved
- \[ = \frac{4 x - 6}{\left(2 x - 3\right)^{2} + 1} + 2 \operatorname{atan}{\left(2 x - 3 \right)} \]algebraSimplify the expression.✓ Proved
- \[ = \frac{4 x - 6}{4 x^{2} - 12 x + 10} + 2 \operatorname{atan}{\left(2 x - 3 \right)} \]algebra simplifyExpand the squared term. Combine like terms in the denominator.✓ Proved
- \[ = \frac{2 x - 3}{2 x^{2} - 6 x + 5} + 2 \operatorname{atan}{\left(2 x - 3 \right)} \]simplifyDivide numerator and denominator by 2.✓ Proved
Answer \( \frac{2 \left(2 x - 3\right)}{\left(2 x - 3\right)^{2} + 1} + 2 \operatorname{atan}{\left(2 x - 3 \right)} \)
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 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (2*x - 3)**2 + 1 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (2*x - 3)**2 + 1 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (2*x - 3)**2 + 1 = 0 undefined where 4*x**2 - 12*x + 10 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*x**2 - 12*x + 10 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*x**2 - 12*x + 10 = 0 undefined where 2*x**2 - 6*x + 5 = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where (2*x - 3)**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: passqwen3.6:27b-mlx: fail (error) — Step 4 applies both the inverse-trig derivative rule and the chain rule (multiplying by the inner derivative 2) simultaneously, violating the one-rule-per-step constraint. Additionally, the label 'inverse-trig' is insufficient to describe the application of the chain rule present in this step.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-21 — Step 4 applies both the inverse-trig derivative rule and the chain rule (multiplying by the inner derivative 2) simultaneously, violating the one-rule-per-step constraint. Additionally, the label 'inverse-trig' is insufficient to describe the application of the chain rule present in this step.gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: pass 2026-09-21gpt-oss:20b: pass 2026-09-21
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.