Home›Calculus 1›Chain rule›Problem 2.308

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

Problem 2.308 · hard

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

\[ \frac{d}{d x} \left(\frac{\ln{\left(\tan^{2}{\left(x - 3 \right)} + 1 \right)}}{2} - \ln{\left(\tan{\left(x - 3 \right)} \right)}\right) \]
derivativeStart with the derivative of the function.✓ Proved
\[ = \frac{d}{d x} \frac{\ln{\left(\tan^{2}{\left(x - 3 \right)} + 1 \right)}}{2} - \frac{d}{d x} \ln{\left(\tan{\left(x - 3 \right)} \right)} \]
sumApply the difference rule.✓ Proved
\[ = \frac{\frac{d}{d x} \ln{\left(\tan^{2}{\left(x - 3 \right)} + 1 \right)}}{2} - \frac{d}{d x} \ln{\left(\tan{\left(x - 3 \right)} \right)} \]
constant-multipleFactor out the constant 1/2.✓ Proved
\[ = - \frac{d}{d x} \ln{\left(\tan{\left(x - 3 \right)} \right)} + \frac{\frac{d}{d x} \left(\tan^{2}{\left(x - 3 \right)} + 1\right)}{2 \left(\tan^{2}{\left(x - 3 \right)} + 1\right)} \]
chainApply the chain rule to the first term.✓ Proved
\[ = - \frac{d}{d x} \ln{\left(\tan{\left(x - 3 \right)} \right)} + \frac{\tan{\left(x - 3 \right)} \frac{d}{d x} \tan{\left(x - 3 \right)}}{\tan^{2}{\left(x - 3 \right)} + 1} \]
chainApply the chain rule to the inner squared term.✓ Proved
\[ = - \frac{d}{d x} \ln{\left(\tan{\left(x - 3 \right)} \right)} + \frac{\tan{\left(x - 3 \right)} \sec^{2}{\left(x - 3 \right)}}{\tan^{2}{\left(x - 3 \right)} + 1} \]
trigDifferentiate tan(x - 3).≈ Checked numerically
\[ = - \frac{\frac{d}{d x} \tan{\left(x - 3 \right)}}{\tan{\left(x - 3 \right)}} + \frac{\tan{\left(x - 3 \right)} \sec^{2}{\left(x - 3 \right)}}{\tan^{2}{\left(x - 3 \right)} + 1} \]
chainApply the chain rule to the second term.✓ Proved
\[ = - \frac{\sec^{2}{\left(x - 3 \right)}}{\tan{\left(x - 3 \right)}} + \frac{\tan{\left(x - 3 \right)} \sec^{2}{\left(x - 3 \right)}}{\tan^{2}{\left(x - 3 \right)} + 1} \]
trig algebraDifferentiate tan(x - 3) in the second term. Simplify the coefficients and terms.≈ Checked numerically
\[ = \left(- \frac{1}{\tan{\left(x - 3 \right)}} + \frac{\tan{\left(x - 3 \right)}}{\tan^{2}{\left(x - 3 \right)} + 1}\right) \sec^{2}{\left(x - 3 \right)} \]
algebraFactor out sec(x - 3)**2.✓ Proved
\[ = \left(- \cot{\left(x - 3 \right)} + \frac{\tan{\left(x - 3 \right)}}{\tan^{2}{\left(x - 3 \right)} + 1}\right) \sec^{2}{\left(x - 3 \right)} \]
rewriteRewrite 1/tan(x - 3) as cot(x - 3).✓ Proved
\[ = - \frac{\sec^{2}{\left(x - 3 \right)}}{\left(\tan^{2}{\left(x - 3 \right)} + 1\right) \tan{\left(x - 3 \right)}} \]
algebra algebra simplify simplifyFind a common denominator. Distribute the denominator. Cancel the tan(x - 3)**2 terms. Final simplified expression.✓ Proved

Answer \( - \frac{1}{\tan{\left(x - 3 \right)}} \)

Lines: 13 proved, 3 checked numerically. 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 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
4	✓ 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 - 3)**2 + 1 = 0
5	✓ 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 - 3)**2 + 1 = 0
6	≈ Checked numerically	sympy 1.14.0	sympy 1.14.0: simplify left (tan(x - 3)2 - sec(x - 3)2 + 1)tan(x - 3)/(tan(x - 3)2 + 1); numeric agreement only, at 24 of 24 sampled points log is undefined for non-positive arguments tan has poles at odd multiples of pi/2 undefined where tan(x - 3)*2 + 1 = 0 sec has poles at odd multiples of pi/2
7	✓ 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 sec has poles at odd multiples of pi/2 undefined where tan(x - 3)**2 + 1 = 0 undefined where tan(x - 3) = 0
8	≈ Checked numerically	sympy 1.14.0	sympy 1.14.0: simplify left (-tan(x - 3)2 + sec(x - 3)2 - 1)/tan(x - 3); 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 - 3) = 0 undefined where tan(x - 3)**2 + 1 = 0
9	✓ 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 - 3) = 0 undefined where tan(x - 3)**2 + 1 = 0
10	✓ 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 - 3) = 0 undefined where tan(x - 3)**2 + 1 = 0
11	✓ 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 - 3) = 0 undefined where tan(x - 3)**2 + 1 = 0 cot has poles at multiples of pi
12	✓ 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 cot has poles at multiples of pi undefined where tan(x - 3)**2 + 1 = 0 undefined where tan(x - 3) = 0
13	✓ 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 - 3) = 0 undefined where tan(x - 3)**2 + 1 = 0
14	✓ 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 - 3) = 0 undefined where tan(x - 3)**2 + 1 = 0
15	✓ 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 - 3) = 0 undefined where tan(x - 3)**2 + 1 = 0
answer	≈ Checked numerically	sympy 1.14.0	sympy 1.14.0: final line against the stated answer: simplify left (tan(x - 3)2 - sec(x - 3)2 + 1)/(tan(x - 3)**3 + tan(x - 3)); numeric agreement only, at 24 of 24 sampled points tan has poles at odd multiples of pi/2 undefined where tan(x - 3) = 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) — The final answer is incorrect. The simplification in step 14 cancels the term (tan(x-3)^2 + 1) in the denominator, but this term equals sec(x-3)^2, which should cancel with the numerator, leaving -1/tan(x-3). The solution fails to perform this cancellation, resulting in an expression that is not equal to the stated answer.

Every verdict on record (12)

qwen3.6:27b-mlx: fail (error) 2026-09-20 — The final answer is incorrect. The simplification in step 14 cancels the term (tan(x-3)^2 + 1) in the denominator, but this term equals sec(x-3)^2, which should cancel with the numerator, leaving -1/tan(x-3). The solution fails to perform this cancellation, resulting in an expression that is not equal to the stated answer.
gpt-oss:20b: pass 2026-09-20
qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The algebraic simplification is sound, and the final result matches the stated answer.
gpt-oss:20b: fail (error) 2026-09-20 — Step 14 incorrectly simplifies sec(x‑3)**2/(tan(x‑3)*(tan(x‑3)**2+1)). Since tan^2+1=sec^2, the fraction should reduce to 1/tan(x‑3), yielding the correct derivative -1/tan(x‑3). The solution therefore contains a mathematical error.
qwen3.6:27b-mlx: fail (misleading) 2026-09-19 — The solution fails to simplify the expression to the stated answer. Step 15 yields -sec(x-3)^2 / (tan(x-3)*(tan(x-3)^2+1)), which simplifies to -1/tan(x-3) using the identity sec^2 = tan^2 + 1, but this final simplification step is missing. The stated answer is correct, but the derivation does not reach it.
deepseek-r1:70b: pass 2026-09-19
gpt-oss:20b: fail (style) 2026-09-19 — The final expression is not simplified to the expected -1/tan(x-3); a missing use of the identity sec^2(x-3)=tan^2(x-3)+1 would reduce the result to the correct form.
qwen3.6:27b-mlx: inconclusive 2026-09-19 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The final answer is mathematically incorrect; the simplification in step 14-15 ignores the identity tan^2(x) + 1 = sec^2(x), which cancels the
deepseek-r1:70b: pass 2026-09-19
gpt-oss:20b: fail (error) 2026-09-19 — The final expression still contains sec(x-3)**2/(tan(x-3)**2+1). Since sec^2(x-3)=tan^2(x-3)+1, this factor cancels, yielding -1/tan(x-3). The solution stops short of this simplification, leaving an incorrect final result.
gpt-oss:20b: fail 2026-09-17 — Step 14’s note claims to cancel tan(x‑3)**2 terms, but the expression only has a -1 numerator; no tan^2 terms are cancelled. The note misleads the student.
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.