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

Problem 2.46 · hard Beautiful

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

\[ \frac{d}{d x} \left(- \ln{\left(\tan^{2}{\left(x - 1 \right)} + 1 \right)} + 2 \ln{\left(\tan{\left(x - 1 \right)} \right)}\right) \]
sumStart with the derivative of the sum.✓ Proved
\[ = \frac{d}{d x} \left(- \ln{\left(\tan^{2}{\left(x - 1 \right)} + 1 \right)}\right) + \frac{d}{d x} 2 \ln{\left(\tan{\left(x - 1 \right)} \right)} \]
constant-multipleDistribute the derivative over the sum.✓ Proved
\[ = \frac{d}{d x} \left(- \ln{\left(\tan^{2}{\left(x - 1 \right)} + 1 \right)}\right) + 2 \frac{d}{d x} \ln{\left(\tan{\left(x - 1 \right)} \right)} \]
constant-multiplePull out the constant coefficient.✓ Proved
\[ = \frac{d}{d x} \left(- \ln{\left(\tan^{2}{\left(x - 1 \right)} + 1 \right)}\right) + \frac{2 \frac{d}{d x} \tan{\left(x - 1 \right)}}{\tan{\left(x - 1 \right)}} \]
logarithmicApply the chain rule for the logarithm.✓ Proved
\[ = \frac{2 \frac{d}{d x} \tan{\left(x - 1 \right)}}{\tan{\left(x - 1 \right)}} - \frac{\frac{d}{d x} \left(\tan^{2}{\left(x - 1 \right)} + 1\right)}{\tan^{2}{\left(x - 1 \right)} + 1} \]
chainApply the chain rule to the first term.✓ Proved
\[ = \frac{2 \frac{d}{d x} \tan{\left(x - 1 \right)}}{\tan{\left(x - 1 \right)}} - \frac{\frac{d}{d x} 1 + \frac{d}{d x} \tan^{2}{\left(x - 1 \right)}}{\tan^{2}{\left(x - 1 \right)} + 1} \]
sumDifferentiate the inner sum.✓ Proved
\[ = \frac{2 \frac{d}{d x} \tan{\left(x - 1 \right)}}{\tan{\left(x - 1 \right)}} - \frac{\frac{d}{d x} \tan^{2}{\left(x - 1 \right)}}{\tan^{2}{\left(x - 1 \right)} + 1} \]
constantThe derivative of a constant is zero.✓ Proved
\[ = \frac{2 \frac{d}{d x} \tan{\left(x - 1 \right)}}{\tan{\left(x - 1 \right)}} - \frac{2 \tan{\left(x - 1 \right)} \frac{d}{d x} \tan{\left(x - 1 \right)}}{\tan^{2}{\left(x - 1 \right)} + 1} \]
powerApply the power rule to the squared term.✓ Proved
\[ = \frac{2 \sec^{2}{\left(x - 1 \right)}}{\tan{\left(x - 1 \right)}} - \frac{2 \tan{\left(x - 1 \right)} \sec^{2}{\left(x - 1 \right)}}{\tan^{2}{\left(x - 1 \right)} + 1} \]
trigDifferentiate the tangent function.≈ Checked numerically
\[ = \left(\frac{2}{\tan{\left(x - 1 \right)}} - \frac{2 \tan{\left(x - 1 \right)}}{\tan^{2}{\left(x - 1 \right)} + 1}\right) \sec^{2}{\left(x - 1 \right)} \]
algebraFactor out the common secant term.✓ Proved
\[ = \frac{2 \sec^{2}{\left(x - 1 \right)}}{\left(\tan^{2}{\left(x - 1 \right)} + 1\right) \tan{\left(x - 1 \right)}} \]
algebra algebra simplify algebraCombine terms using a common denominator. Expand the numerator. Simplify the numerator by canceling terms. Multiply the secant term into the numerator.✓ Proved
\[ = \frac{2}{\left(\tan^{2}{\left(x - 1 \right)} + 1\right) \cos^{2}{\left(x - 1 \right)} \tan{\left(x - 1 \right)}} \]
rewrite algebraRewrite secant in terms of cosine. Combine the fractions.✓ Proved
\[ = \frac{2}{\left(\tan^{2}{\left(x - 1 \right)} + 1\right) \sin{\left(x - 1 \right)} \cos{\left(x - 1 \right)}} \]
rewrite algebraRewrite tangent in terms of sine and cosine. Cancel one cosine term from the denominator.✓ Proved
\[ = \frac{2}{\sin{\left(x - 1 \right)} \cos{\left(x - 1 \right)} \sec^{2}{\left(x - 1 \right)}} \]
rewriteRewrite the term (tan(x-1)**2 + 1) as sec(x-1)**2.≈ Checked numerically
\[ = \frac{2 \cos{\left(x - 1 \right)}}{\sin{\left(x - 1 \right)}} \]
rewrite algebra simplifyRewrite secant squared as 1/cos squared. Move the cosine squared term to the numerator. Simplify the fraction by canceling one cosine term.✓ Proved
\[ = 2 \cot{\left(x - 1 \right)} \]
trigRewrite the ratio as the cotangent function.✓ Proved

Answer \( \frac{2}{\tan{\left(x - 1 \right)}} \)

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.

Lines: 22 proved, 2 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 - 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 - 1) = 0 undefined where tan(x - 1)**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 - 1)**2 + 1 = 0 undefined where tan(x - 1) = 0
7	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 undefined where tan(x - 1)**2 + 1 = 0 undefined where tan(x - 1) = 0
8	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 undefined where tan(x - 1)**2 + 1 = 0 undefined where tan(x - 1) = 0
9	≈ Checked numerically	sympy 1.14.0	sympy 1.14.0: simplify left (2tan(x - 1)2 - 2sec(x - 1)2 + 2)/(tan(x - 1)3 + tan(x - 1)); numeric agreement only, at 24 of 24 sampled points tan has poles at odd multiples of pi/2 undefined where tan(x - 1) = 0 undefined where tan(x - 1)**2 + 1 = 0 sec has poles at odd multiples of pi/2
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 - 1) = 0 undefined where tan(x - 1)**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 - 1) = 0 undefined where tan(x - 1)**2 + 1 = 0
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 undefined where tan(x - 1)**2 + 1 = 0 undefined where tan(x - 1) = 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 - 1)**2 + 1 = 0 undefined where tan(x - 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 - 1)**2 + 1 = 0 undefined where tan(x - 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 - 1)**2 + 1 = 0 undefined where tan(x - 1) = 0 undefined where cos(x - 1) = 0
16	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 undefined where tan(x - 1)**2 + 1 = 0 undefined where cos(x - 1) = 0 undefined where tan(x - 1) = 0
17	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 undefined where tan(x - 1)**2 + 1 = 0 undefined where cos(x - 1) = 0 undefined where tan(x - 1) = 0 undefined where sin(x - 1) = 0
18	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 undefined where tan(x - 1)**2 + 1 = 0 undefined where cos(x - 1) = 0 undefined where sin(x - 1) = 0
19	≈ Checked numerically	sympy 1.14.0	sympy 1.14.0: simplify left (-2tan(x - 1)2 + 2sec(x - 1)2 - 2)/((tan(x - 1)2 + 1)tan(x - 1)); numeric agreement only, at 24 of 24 sampled points tan has poles at odd multiples of pi/2 undefined where tan(x - 1)*2 + 1 = 0 undefined where cos(x - 1) = 0 undefined where sin(x - 1) = 0 sec has poles at odd multiples of pi/2 undefined where sec(x - 1) = 0
20	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 sec has poles at odd multiples of pi/2 undefined where cos(x - 1) = 0 undefined where sec(x - 1) = 0 undefined where sin(x - 1) = 0
21	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where sin(x - 1) = 0
22	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where sin(x - 1) = 0
23	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where sin(x - 1) = 0 cot has poles at multiples of pi
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 undefined where tan(x - 1) = 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 2*cot(x - 1) is mathematically incorrect; the correct derivative is 2*csc(2*(x - 1)). The error occurs in step 21, where the algebraic manipulation of the denominator term (1/cos^2) is handled incorrectly, leading to an incorrect numerator factor of cos^2 instead of the correct simplification path.

Every verdict on record (12)

qwen3.6:27b-mlx: fail (error) 2026-09-20 — The final answer 2*cot(x - 1) is mathematically incorrect; the correct derivative is 2*csc(2*(x - 1)). The error occurs in step 21, where the algebraic manipulation of the denominator term (1/cos^2) is handled incorrectly, leading to an incorrect numerator factor of cos^2 instead of the correct simplification path.
gpt-oss:20b: pass 2026-09-20
qwen3.6:27b-mlx: fail (style) 2026-09-20 — Step 2 is labeled 'constant-multiple' but performs the distribution of the derivative over a sum, which should be labeled 'sum'. Step 3 is labeled 'constant-multiple' and correctly pulls out the constant 2, but the previous step's mislabeling violates the contract that each step must name the rule it applied.
gpt-oss:20b: pass 2026-09-20
qwen3.6:27b-mlx: fail (error) 2026-09-19 — Step 2 is labeled 'constant-multiple' but performs the linearity of the derivative over a sum (splitting the derivative of a sum into a sum of derivatives), which should be labeled 'sum'. Step 4 is labeled 'logarithmic' but applies the chain rule to the logarithm; while 'logarithmic' is in the vocabulary, the step explicitly computes the derivative of the inner function via the chain rule structure, making 'chain' a more precise label for the operation performed on the composite function, or 'logarithmic' if referring strictly to the outer derivative. However, the primary defect is Step 2 mislabeling the sum rule as constant-multiple.
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 is labeled 'constant-multiple' but performs the distribution of the derivative over a sum, which should be labeled 'sum'. Step 3 is labeled 'constant-multiple' but the note says 'Pull out the constant coefficient', which is correct for the label, but Step 2's label is incorrect for the operation performed (splitting the sum). The contract requires each step to name its rule correctly. Step 2 applies the sum rule (linearity of differentiation), not the constant multiple rule. This is a labeling defect.
deepseek-r1:70b: pass 2026-09-19
gpt-oss:20b: pass 2026-09-19
gpt-oss:20b: fail 2026-09-17 — Step 2 incorrectly labels the operation as "constant‑multiple" when it is actually the sum rule; the note also misstates the rule applied.
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.