Derivative of \( \displaystyle \frac{3 \ln{\left(\tan^{2}{\left(x + 2 \right)} + 1 \right)}}{2} \)

Problem 2.88 · hard Beautiful

Differentiate \( \displaystyle f(x) = \frac{3 \ln{\left(\tan^{2}{\left(x + 2 \right)} + 1 \right)}}{2} \).

\[ \frac{d}{d x} \frac{3 \ln{\left(\tan^{2}{\left(x + 2 \right)} + 1 \right)}}{2} \]
derivativeStart with the derivative of the function.✓ Proved
\[ = \frac{3 \frac{d}{d x} \ln{\left(\tan^{2}{\left(x + 2 \right)} + 1 \right)}}{2} \]
constant-multiple✓ Proved
\[ = \frac{3 \frac{d}{d x} \left(\tan^{2}{\left(x + 2 \right)} + 1\right)}{2 \left(\tan^{2}{\left(x + 2 \right)} + 1\right)} \]
logarithmicApply the chain rule for the natural logarithm.✓ Proved
\[ = \frac{3 \left(\frac{d}{d x} 1 + \frac{d}{d x} \tan^{2}{\left(x + 2 \right)}\right)}{2 \left(\tan^{2}{\left(x + 2 \right)} + 1\right)} \]
sumApply the sum rule to the inner expression.✓ Proved
\[ = \frac{3 \frac{d}{d x} \tan^{2}{\left(x + 2 \right)}}{2 \left(\tan^{2}{\left(x + 2 \right)} + 1\right)} \]
constantThe derivative of a constant is zero.✓ Proved
\[ = \frac{3 \tan{\left(x + 2 \right)} \frac{d}{d x} \tan{\left(x + 2 \right)}}{\tan^{2}{\left(x + 2 \right)} + 1} \]
powerApply the power rule and chain rule.✓ Proved
\[ = \frac{3 \tan{\left(x + 2 \right)} \sec^{2}{\left(x + 2 \right)}}{\tan^{2}{\left(x + 2 \right)} + 1} \]
derivative algebraThe derivative of tan(x + 2) is sec(x + 2)**2. Simplify the coefficients and multiply the terms.≈ Checked numerically
\[ = 3 \tan{\left(x + 2 \right)} \]
rewrite simplifyUse the identity 1 + tan(x + 2)**2 = sec(x + 2)**2. Cancel the common sec(x + 2)**2 term.≈ Checked numerically

Answer \( 3 \tan{\left(x + 2 \right)} \)

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.

✓ Nihil obstat Lines: 9 proved, 2 checked numerically. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.

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 log is undefined for non-positive arguments tan has poles at odd multiples of pi/2
3	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments tan has poles at odd multiples of pi/2 undefined where tan(x + 2)**2 + 1 = 0
4	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 undefined where tan(x + 2)**2 + 1 = 0
5	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 undefined where tan(x + 2)**2 + 1 = 0
6	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 undefined where tan(x + 2)**2 + 1 = 0
7	≈ Checked numerically	sympy 1.14.0	sympy 1.14.0: simplify left 3(tan(x + 2)2 - sec(x + 2)2 + 1)tan(x + 2)/(tan(x + 2)2 + 1); numeric agreement only, at 24 of 24 sampled points tan has poles at odd multiples of pi/2 undefined where tan(x + 2)2 + 1 = 0 sec has poles at odd multiples of pi/2
8	✓ 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 undefined where tan(x + 2)**2 + 1 = 0
9	≈ Checked numerically	sympy 1.14.0	sympy 1.14.0: simplify left 3(-tan(x + 2)2 + sec(x + 2)2 - 1)tan(x + 2)/(tan(x + 2)2 + 1); numeric agreement only, at 24 of 24 sampled points tan has poles at odd multiples of pi/2 sec has poles at odd multiples of pi/2 undefined where tan(x + 2)2 + 1 = 0
10	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 tan 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 tan has poles at odd multiples of pi/2
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 (style) — Step 6 applies both the power rule and the chain rule simultaneously, violating the constraint that each step must change only one thing. It should be split into a power rule step followed by a chain rule step.

Every verdict on record (12)

qwen3.6:27b-mlx: fail (style) 2026-09-20 — Step 6 applies both the power rule and the chain rule simultaneously, violating the constraint that each step must change only one thing. It should be split into a power rule step followed by a chain rule step.
gpt-oss:20b: pass 2026-09-20
qwen3.6:27b-mlx: fail (style) 2026-09-20 — Step 6 applies both the power rule and the chain rule simultaneously, violating the one-rule-per-step constraint. Step 3's note incorrectly identifies the operation as 'the chain rule for the natural logarithm' when the label is 'logarithmic' and the step explicitly separates the outer derivative from the inner derivative.
gpt-oss:20b: pass 2026-09-20
qwen3.6:27b-mlx: fail (style) 2026-09-19 — Step 6 applies both the power rule and the chain rule simultaneously, violating the contract that each step must change only one thing using a single rule. It should be split into a power rule step followed by a chain rule step.
deepseek-r1:70b: pass 2026-09-19
gpt-oss:20b: pass 2026-09-19
qwen3.6:27b-mlx: fail (style) 2026-09-19 — Step 6 applies both the power rule and the chain rule but is labeled only 'power', violating the one-rule-per-step constraint.
deepseek-r1:70b: pass 2026-09-19
gpt-oss:20b: pass 2026-09-19
gpt-oss:20b: pass 2026-09-17
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.