Derivative of \( \displaystyle \frac{3 \tan{\left(2 x \right)}}{2} \)
Problem 2.671 · easy
Differentiate \( \displaystyle f(x) = \frac{3 \tan{\left(2 x \right)}}{2} \).
- \[ \frac{d}{d x} \frac{3 \tan{\left(2 x \right)}}{2} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{3 \frac{d}{d x} \tan{\left(2 x \right)}}{2} \]constant-multiple rewrite simplifyPull out the constant factor. This step is not applicable here as tan(2x) is not a base-exponent form, but we apply the chain rule directly. The previous rewrite was unnecessary; we proceed with the chain rule.✓ Proved
- \[ = \frac{3 \sec^{2}{\left(2 x \right)} \frac{d}{d x} 2 x}{2} \]chainApply the chain rule to tan(2*x).✓ Proved
- \[ = 3 \sec^{2}{\left(2 x \right)} \]derivative algebraDifferentiate the inner function 2*x. Multiply the constants.✓ Proved
Answer \( \frac{3}{\cos^{2}{\left(2 x \right)}} \)
Mind the domain. The answer is also defined at points where f(x) is not. Substituting there gives a number that is not a slope of f.
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 tan has poles at odd multiples of pi/2 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 sec has poles at odd multiples of pi/2 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 sec has poles at odd multiples of pi/2 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 sec has poles at odd multiples of pi/2 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where cos(2*x) = 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 (style) — [domain objection, downgraded to style] Step 3 applies the rule 'rewrite' to transform tan(2*x) into exp(log(tan(2*x))), which is mathematically invalid for all real x because the logarithm of a negative number (where tan is negative) is undefined in the real domain. Furthermore, the note admits the step is not applicable, making it a logical error in the derivation.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (style) 2026-09-21 — [domain objection, downgraded to style] Step 3 applies the rule 'rewrite' to transform tan(2*x) into exp(log(tan(2*x))), which is mathematically invalid for all real x because the logarithm of a negative number (where tan is negative) is undefined in the real domain. Furthermore, the note admits the step is not applicable, making it a logical error in the derivation.gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: fail (error) 2026-09-21 — Step 3 applies a 'rewrite' rule to transform tan(2*x) into exp(log(tan(2*x))), which is algebraically valid but logically nonsensical in this context as the note admits it is not applicable and the step is immediately undone in Step 4. This violates the contract that steps must be legitimate applications of rules towards the solution, not arbitrary and self-correcting detours.gpt-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.