Home›Calculus 1›Derivatives of trigonometric functions›Problem 2.671

Derivative of \( \displaystyle \frac{3 \tan{\left(2 x \right)}}{2} \)

Problem 2.671 · easy Video

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 \( \frac{3}{\cos^{2}{\left(2 x \right)}} \)

Top: the graph of f = 3tan(2x)/2. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 3tan(2x)² + 3.

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. A reviewer's objection to the explanation stands; see the receipt. 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 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: pass
qwen3.6:27b-mlx: fail (error) — Step 3 applies a 'rewrite' rule that transforms tan(2*x) into exp(log(tan(2*x))), which is not a standard or valid simplification for differentiation and introduces unnecessary complexity. Step 4 then undoes this with 'simplify', making the sequence logically incoherent and pedagogically harmful.

Senior review claude-sonnet-5-5, 2026-10-11: fail — The math is correct and the final answer matches. However, lines 3-4 are a spurious detour: tan(2x) is rewritten as exp(log(tan(2x))) and then reverted. The line 3 note says the step is 'not applicable', which contradicts its own rewrite label and makes the derivation incoherent for a student.

qwen3.6:27b-mlx: uphold — Line 3 is a mathematically true identity (exp(log(tan)) = tan), but the note contradicts the step. It says the step 'is not applicable' while labelling it a rewrite, then line 4 undoes it as 'unnecessary'. This is a pointless detour with self-contradictory exposition that a student would find confusing, though no math is false.

Every verdict on record (7)

claude-sonnet-5-5: fail (misleading) 2026-10-11 — The math is correct and the final answer matches. However, lines 3-4 are a spurious detour: tan(2x) is rewritten as exp(log(tan(2x))) and then reverted. The line 3 note says the step is 'not applicable', which contradicts its own rewrite label and makes the derivation incoherent for a student.
qwen3.6:27b-mlx: fail (error) 2026-09-26 — Step 3 applies a 'rewrite' rule that transforms tan(2*x) into exp(log(tan(2*x))), which is not a standard or valid simplification for differentiation and introduces unnecessary complexity. Step 4 then undoes this with 'simplify', making the sequence logically incoherent and pedagogically harmful.
gpt-oss:20b: pass 2026-09-26
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 2026-09-21
qwen3.6:27b-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.