Home›Calculus 1›Derivatives of trigonometric functions›Problem 2.1251

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

Problem 2.1251 · hard Beautiful

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 sum 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-multiplePull out the constant factor.✓ 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
\[ = \frac{d}{d x} \ln{\left(\tan{\left(x - 3 \right)} \right)} - \frac{\frac{d}{d x} 1 + \frac{d}{d x} \tan^{2}{\left(x - 3 \right)}}{2 \left(\tan^{2}{\left(x - 3 \right)} + 1\right)} \]
sumApply the sum rule inside the derivative.✓ Proved
\[ = \frac{d}{d x} \ln{\left(\tan{\left(x - 3 \right)} \right)} - \frac{\frac{d}{d x} \tan^{2}{\left(x - 3 \right)}}{2 \left(\tan^{2}{\left(x - 3 \right)} + 1\right)} \]
constantThe derivative of a constant is zero.✓ Proved
\[ = \frac{d}{d x} \ln{\left(\tan{\left(x - 3 \right)} \right)} - \frac{\tan{\left(x - 3 \right)} \frac{d}{d x} \tan{\left(x - 3 \right)}}{\tan^{2}{\left(x - 3 \right)} + 1} \]
powerApply the power rule to tan(x-3)**2.✓ Proved
\[ = \frac{d}{d x} \ln{\left(\tan{\left(x - 3 \right)} \right)} - \frac{\tan{\left(x - 3 \right)} \frac{d}{d x} \left(x - 3\right)}{\left(\tan^{2}{\left(x - 3 \right)} + 1\right) \cos^{2}{\left(x - 3 \right)}} \]
chainApply the chain rule to tan(x-3).≈ Checked numerically
\[ = \frac{d}{d x} \ln{\left(\tan{\left(x - 3 \right)} \right)} - \frac{\tan{\left(x - 3 \right)}}{\left(\tan^{2}{\left(x - 3 \right)} + 1\right) \cos^{2}{\left(x - 3 \right)}} \]
derivative algebra algebraThe derivative of x-3 is 1. Multiply the terms together. Simplify the fraction by canceling 2.✓ Proved
\[ = \frac{d}{d x} \ln{\left(\tan{\left(x - 3 \right)} \right)} - \frac{\tan{\left(x - 3 \right)}}{\cos^{2}{\left(x - 3 \right)} \tan^{2}{\left(x - 3 \right)} + \cos^{2}{\left(x - 3 \right)}} \]
algebraDistribute the denominator term.✓ Proved
\[ = \frac{d}{d x} \ln{\left(\tan{\left(x - 3 \right)} \right)} - \frac{\tan{\left(x - 3 \right)}}{\sin^{2}{\left(x - 3 \right)} + \cos^{2}{\left(x - 3 \right)}} \]
algebraUse the identity tan^2 + 1 = sec^2, or simplify via sin/cos.✓ Proved
\[ = - \tan{\left(x - 3 \right)} + \frac{d}{d x} \ln{\left(\tan{\left(x - 3 \right)} \right)} \]
simplifyUse the Pythagorean identity sin^2 + cos^2 = 1.✓ Proved
\[ = - \tan{\left(x - 3 \right)} + \frac{\frac{d}{d x} \tan{\left(x - 3 \right)}}{\tan{\left(x - 3 \right)}} \]
chainApply the chain rule to the second term.✓ Proved
\[ = - \tan{\left(x - 3 \right)} + \frac{1}{\cos^{2}{\left(x - 3 \right)} \tan{\left(x - 3 \right)}} \]
chain algebraApply the chain rule to tan(x-3). Simplify the product of fractions.≈ Checked numerically
\[ = - \tan{\left(x - 3 \right)} + \frac{1}{\sin{\left(x - 3 \right)} \cos{\left(x - 3 \right)}} \]
algebra algebraUse tan(u) = sin(u)/cos(u) to simplify the denominator. Introduce a 2 to facilitate the double angle identity.✓ Proved
\[ = - \tan{\left(x - 3 \right)} + \frac{2}{\sin{\left(2 x - 6 \right)}} \]
algebraApply the double angle identity sin(2u) = 2sin(u)cos(u).✓ Proved
\[ = - \tan{\left(x - 3 \right)} + 2 \csc{\left(2 x - 6 \right)} \]
rewrite algebraRewrite 1/sin(u) as csc(u). Distribute the 2 inside the argument.✓ Proved

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

Lines: 20 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	✓ 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
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 undefined where tan(x - 3)**2 + 1 = 0
8	≈ Checked numerically	sympy 1.14.0	sympy 1.14.0: simplify left (-(tan(x - 3)*2 + 1)cos(x - 3)*2 + 1)tan(x - 3)/((tan(x - 3)*2 + 1)cos(x - 3)2); 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 undefined where cos(x - 3) = 0
9	✓ 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 cos(x - 3) = 0 undefined where tan(x - 3)**2 + 1 = 0
10	✓ 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 cos(x - 3) = 0 undefined where tan(x - 3)**2 + 1 = 0
11	✓ 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 cos(x - 3) = 0 undefined where tan(x - 3)**2 + 1 = 0
12	✓ 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 cos(x - 3) = 0 undefined where tan(x - 3)2 + 1 = 0 undefined where cos(x - 3)2tan(x - 3)2 + cos(x - 3)*2 = 0
13	✓ 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 cos(x - 3)*2tan(x - 3)2 + cos(x - 3)2 = 0 undefined where sin(x - 3)2 + cos(x - 3)2 = 0
14	✓ 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 sin(x - 3)2 + cos(x - 3)2 = 0
15	✓ 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) = 0
16	≈ Checked numerically	sympy 1.14.0	sympy 1.14.0: simplify left tan(x - 3) + 1/tan(x - 3) - 1/(cos(x - 3)*2tan(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 undefined where cos(x - 3) = 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 - 3) = 0 undefined where cos(x - 3) = 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 - 3) = 0 undefined where cos(x - 3) = 0 undefined where sin(x - 3) = 0
19	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 undefined where sin(x - 3) = 0 undefined where cos(x - 3) = 0
20	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 undefined where sin(x - 3) = 0 undefined where cos(x - 3) = 0 undefined where sin(2*x - 6) = 0
21	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 undefined where sin(2*x - 6) = 0 csc has poles at multiples of pi
22	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 csc has poles at multiples of pi
answer	≈ Checked numerically	sympy 1.14.0	sympy 1.14.0: final line against the stated answer: simplify left -tan(x - 3) + 2csc(2x - 6) - 1/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: fail (error) — Several steps apply multiple rules at once and contain incorrect algebraic manipulations. For example, step 12 distributes the denominator incorrectly, step 13 applies an identity to a wrong expression, and the final result diverges from the correct derivative 1/tan(x‑3).
qwen3.6:27b-mlx: pass — The solution correctly applies differentiation rules and algebraic simplifications. Each step isolates a single rule application, and the labels accurately reflect the operations performed.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies differentiation rules and algebraic simplifications. Each step isolates a single rule application, and the labels accurately reflect the operations performed.
gpt-oss:20b: fail (error) 2026-09-29 — Several steps apply multiple rules at once and contain incorrect algebraic manipulations. For example, step 12 distributes the denominator incorrectly, step 13 applies an identity to a wrong expression, and the final result diverges from the correct derivative 1/tan(x‑3).
qwen3.6:27b-mlx: inconclusive 2026-09-29 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "Step 15 applies the chain rule to the second term but fails to differentiate the outer function log(u) into 1/u * u', instead leaving the derivative
gpt-oss:20b: fail (error) 2026-09-29 — Step 12 incorrectly distributes the denominator, turning a product into a sum. Step 13 then misapplies the identity tan^2+1=sec^2, leading to an invalid simplification. These errors invalidate the entire differentiation chain.

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 2026-09-29 with SymPy 1.14.0.