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

Problem 2.26 · medium Mental math

Differentiate \( \displaystyle f(x) = \tan{\left(3 x \right)} \).

\[ \frac{d}{d x} \tan{\left(3 x \right)} \]
derivativeStart with the derivative of the function.✓ Proved
\[ = \sec^{2}{\left(3 x \right)} \frac{d}{d x} 3 x \]
trigApply the derivative rule for tangent.✓ Proved
\[ = 3 \sec^{2}{\left(3 x \right)} \]
derivative algebraDifferentiate the inner function 3*x. Rearrange the terms.✓ Proved

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

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 sec has poles at odd multiples of pi/2
3	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 sec has poles at odd multiples of pi/2
4	✓ 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(3*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
deepseek-r1:70b: pass
qwen3.6:27b-mlx: fail (error) — Step 2 applies the chain rule to differentiate the composite function, but is labeled 'trig'. The label 'trig' is reserved for trigonometric identities or simplifications, not for the differentiation rule for tangent which requires the chain rule.

Every verdict on record (12)

qwen3.6:27b-mlx: fail (error) 2026-09-20 — Step 2 applies the chain rule to differentiate the composite function, but is labeled 'trig'. The label 'trig' is reserved for trigonometric identities or simplifications, not for the differentiation rule for tangent which requires the chain rule.
gpt-oss:20b: pass 2026-09-20
qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies the chain rule in two steps (differentiating the outer tangent function, then the inner linear function) and uses valid labels from the vocabulary.
gpt-oss:20b: fail (error) 2026-09-20 — Step 2 applies both the tangent derivative rule and the derivative of the inner function 3*x in one line, violating the rule that each step must change only one thing. The label "trig" is also incomplete for that step.
qwen3.6:27b-mlx: fail (error) 2026-09-19 — Step 2 applies both the chain rule and the derivative of tangent simultaneously, violating the one-rule-per-step constraint. It should be split into a chain rule step introducing the inner derivative, followed by a trig/derivative step for tan.
deepseek-r1:70b: pass 2026-09-19
gpt-oss:20b: pass 2026-09-19
qwen3.6:27b-mlx: fail (error) 2026-09-19 — Step 2 applies the chain rule to differentiate tan(3*x), but is labeled 'trig'. The label 'trig' is reserved for trigonometric identities or simplifications, not for the application of the chain rule during differentiation.
deepseek-r1:70b: pass 2026-09-19
gpt-oss:20b: pass 2026-09-19
gpt-oss:20b: fail 2026-09-17 — The solution omits the domain restriction cos(3*x)≠0, which is necessary for the derivative to exist.
deepseek-r1:70b: pass 2026-09-17

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.