Derivative of \( \displaystyle - \frac{\tan{\left(4 x \right)}}{4} \)

Reviewers

• gpt-oss:20b: pass
• deepseek-r1:70b: pass
• qwen3.6:27b-mlx: fail (error) — The stated answer is -tan(4*x)**2 - 1, but the final step of the solution is -sec(4*x)**2. These expressions are not algebraically equal (sec^2(u) = tan^2(u) + 1, so -sec^2(u) = -tan^2(u) - 1, but the stated answer has a sign error relative to the derived result or is simply incorrect as a simplification of the derivative). Specifically, the derivative of -tan(4x)/4 is -sec^2(4x). The stated answer claims the result is -tan^2(4x) - 1, which equals -sec^2(4x). Wait, -tan^2 - 1 = -(tan^2 + 1) = -sec^2. So the stated answer IS algebraically equal to the final step. Let me re-read the prompt constraints. 'A computer algebra system has already confirmed that consecutive lines are algebraically equal'. Okay, so I don't need to check algebra. I need to check the steps against the contract. Let's re-evaluate. Step 1: Derivative(-tan(4*x)/4, x). Label: derivative. This is just stating the problem. Acceptable. Step 2: -1/4 * Derivative(tan(4*x), x). Label: constant. This pulls out the constant factor -1/4. This is `constant-multiple` rule effectively, but `constant` is in the vocabulary? The vocabulary has `constant` and `constant-multiple`. The rule applied is pulling out a constant multiplier. The label `constant` is ambiguous. Usually `constant` refers to the derivative of a constant being 0. `constant-multiple` is the rule d/dx [c f(x)] = c f'(x). The step applies `constant-multiple`. The label used is `constant`. Is `constant` a valid label for this? The prompt says: 'Each step names its rule from this fixed vocabulary... A label is correct if it names a rule the step legitimately applied.' It does not say `constant` is a synonym for `constant-multiple`. In standard calculus terminology, 'Constant Rule' is d/dx(c)=0. 'Constant Multiple Rule' is d/dx(cf)=cf'. The step is clearly the Constant Multiple Rule. Using the label `constant` is likely a defect because it names the wrong rule (derivative of a constant vs constant multiple). However, some systems might use `constant` loosely. Let's look at the other steps. Step 3: -1/4 * sec(4*x)**2 * Derivative(4*x, x). Label: chain. This applies the chain rule to tan(4x). Correct. Step 4: -1/4 * sec(4*x)**2 * 4. Label: derivative. This differentiates 4x to get 4. Correct. Step 5: -sec(4*x)**2. Label: algebra. Simplifies -1/4 * 4 to -1. Correct. The main issue is Step 2's label. The rule applied is the constant multiple rule. The label `constant` typically refers to the derivative of a constant function. The vocabulary includes `constant-multiple`. Therefore, using `constant` instead of `constant-multiple` is a labeling defect. Is it a 'style' or 'error'? The prompt says 'A label is correct if it names a rule the step legitimately applied.' Did the step apply the 'constant' rule? No, it applied the 'constant-multiple' rule. So the label is incorrect. This is a defect. Is it 'style' or 'error'? 'Style' is for 'naming and granularity complaints'. This is a naming complaint. So severity 'style'. Wait, let's look closer at the stated answer vs the final step. The prompt says 'A computer algebra system has already confirmed that consecutive lines are algebraically equal'. This implies I should trust the algebra. The final line is `-sec(4*x)**2`. The stated answer is `-tan(4*x)**2 - 1`. These are equal. The solution ends at step 5 with `-sec(4*x)**2`. It does not explicitly show the step to convert to the stated answer format, but the task is to differentiate, and the final expression is a valid form of the derivative. The prompt asks to review the *steps*. The steps lead to `-sec(4*x)**2`. The stated answer is equivalent. This is usually acceptable unless the contract requires the final form to match exactly. The prompt doesn't specify the final form must match the stated answer, just that the solution is judged against the contract. The contract is about the steps. Let's re-read the label constraint carefully. 'constant-multiple' is in the list. 'constant' is in the list. Step 2 pulls out a constant. This is the constant multiple rule. Labeling it 'constant' is incorrect because 'constant' usually means d/dx(c) = 0. This is a mislabeling. Severity: style. Is there any other defect? Step 1: `Derivative(-tan(4*x)/4, x)` labeled `derivative`. This is just the setup. It's not really applying a rule, it's stating the operation. But often the first step is just the problem statement. If it's considered a step, it's trivial. Step 2: `constant` label. As discussed, should be `constant-multiple`. Step 3: `chain` label. Correct. Step 4: `derivative` label. Correct. Step 5: `algebra` label. Correct. So the only defect is the label in Step 2. It should be `constant-multiple`. Using `constant` is a naming error. Since `constant-multiple` exists in the vocabulary, using `constant` is a failure to use the precise label. This falls under 'style' as per 'naming ... complaints go here'. However, sometimes 'constant' is used to mean 'constant multiple' in sloppy contexts. But the prompt is adversarial and provides a specific vocabulary including `constant-multiple`. This suggests precision is required. Therefore, `constant` is a wrong label for the rule applied. Verdict: fail. Severity: style. Notes: Step 2 applies the constant multiple rule but is labeled 'constant', which typically refers to the derivative of a constant being zero; the vocabulary includes 'constant-multiple' which is the correct label.

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.