Derivative of \( \displaystyle \left(2 x - 3\right) \operatorname{atan}{\left(2 x - 3 \right)} \)
Problem 2.650 · medium Video
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 \( \frac{2 \left(2 x - 3\right)}{\left(2 x - 3\right)^{2} + 1} + 2 \operatorname{atan}{\left(2 x - 3 \right)} \)
✓ 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 |
| 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: fail (style) — Step 1 lacks a proper rule label from the allowed vocabulary. It should be labeled "derivative" (or another appropriate rule) rather than left unlabelled.qwen3.6:27b-mlx: fail (error) — Step 4 applies the chain rule (multiplying by the derivative of the inner function 2) in addition to the inverse-trig derivative rule, violating the one-rule-per-step constraint. Step 3 also applies the chain rule implicitly when differentiating the inner function of the arctan term is not yet complete, but the primary defect is Step 4 combining inverse-trig and chain rules.
Senior review claude-sonnet-5-5, 2026-10-11: pass — Every line is an equivalent form of the one before, SymPy proved each step, and the final line equals the stated answer. The rule labels are legitimate and no step changes more than one thing in a way that counts as a defect.
gpt-oss:20b: dismiss — Line 1 just restates the problem, which SymPy proved, so it needs no rule label and nothing in it is false.qwen3.6:27b-mlx: dismiss — Differentiating atan(2x-3) and evaluating the inner derivative 2 is a single chain/inverse-trig step under the contract, and 'inverse-trig' names a rule that was actually applied. Line 3 only differentiated 2x-3 and did not touch the atan term.
Every verdict on record (7)
claude-sonnet-5-5: pass 2026-10-11 — Every line is an equivalent form of the one before, SymPy proved each step, and the final line equals the stated answer. The rule labels are legitimate and no step changes more than one thing in a way that counts as a defect.qwen3.6:27b-mlx: fail (error) 2026-09-26 — Step 4 applies the chain rule (multiplying by the derivative of the inner function 2) in addition to the inverse-trig derivative rule, violating the one-rule-per-step constraint. Step 3 also applies the chain rule implicitly when differentiating the inner function of the arctan term is not yet complete, but the primary defect is Step 4 combining inverse-trig and chain rules.gpt-oss:20b: fail (style) 2026-09-26 — Step 1 lacks a proper rule label from the allowed vocabulary. It should be labeled "derivative" (or another appropriate rule) rather than left unlabelled.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.