Home›Calculus 1›Derivatives of trigonometric functions›Problem 2.871

Derivative of \( \displaystyle \frac{\ln{\left(\tan^{2}{\left(2 x - 3 \right)} + 1 \right)}}{4} \)

Problem 2.871 · hard Beautiful

Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\tan^{2}{\left(2 x - 3 \right)} + 1 \right)}}{4} \).

\[ \frac{d}{d x} \frac{\ln{\left(\tan^{2}{\left(2 x - 3 \right)} + 1 \right)}}{4} \]
derivativeStart with the derivative of the function.✓ Proved
\[ = \frac{\frac{d}{d x} \ln{\left(\tan^{2}{\left(2 x - 3 \right)} + 1 \right)}}{4} \]
constant-multiplePull out the constant factor.✓ Proved
\[ = \frac{\frac{d}{d x} \left(\tan^{2}{\left(2 x - 3 \right)} + 1\right)}{4 \left(\tan^{2}{\left(2 x - 3 \right)} + 1\right)} \]
chainApply the chain rule to the logarithm.✓ Proved
\[ = \frac{\frac{d}{d x} 1 + \frac{d}{d x} \tan^{2}{\left(2 x - 3 \right)}}{4 \left(\tan^{2}{\left(2 x - 3 \right)} + 1\right)} \]
sumApply the sum rule to the inner expression.✓ Proved
\[ = \frac{\frac{d}{d x} \tan^{2}{\left(2 x - 3 \right)}}{4 \left(\tan^{2}{\left(2 x - 3 \right)} + 1\right)} \]
constantThe derivative of a constant is zero.✓ Proved
\[ = \frac{\tan{\left(2 x - 3 \right)} \frac{d}{d x} \tan{\left(2 x - 3 \right)}}{2 \left(\tan^{2}{\left(2 x - 3 \right)} + 1\right)} \]
powerApply the power rule to the squared tangent term.✓ Proved
\[ = \frac{\tan{\left(2 x - 3 \right)} \sec^{2}{\left(2 x - 3 \right)} \frac{d}{d x} \left(2 x - 3\right)}{2 \left(\tan^{2}{\left(2 x - 3 \right)} + 1\right)} \]
trigDifferentiate the tangent function.≈ Checked numerically
\[ = \frac{\tan{\left(2 x - 3 \right)} \sec^{2}{\left(2 x - 3 \right)}}{\tan^{2}{\left(2 x - 3 \right)} + 1} \]
chain algebra algebraDifferentiate the linear inner function. Multiply the constants. Simplify the coefficient 4/4 to 1.✓ Proved
\[ = \tan{\left(2 x - 3 \right)} \]
rewrite simplifySubstitute sec(x)**2 with 1 + tan(x)**2. Cancel the common term (tan(2*x - 3)**2 + 1).≈ Checked numerically

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

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: 11 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(2x - 3)*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(2x - 3)*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(2x - 3)*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(2x - 3)*2 + 1 = 0
7	≈ Checked numerically	sympy 1.14.0	sympy 1.14.0: simplify left (tan(2x - 3)2 - sec(2x - 3)*2 + 1)tan(2x - 3)/(tan(2x - 3)*2 + 1); numeric agreement only, at 24 of 24 sampled points tan has poles at odd multiples of pi/2 undefined where tan(2x - 3)**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(2x - 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(2x - 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(2x - 3)*2 + 1 = 0
11	≈ Checked numerically	sympy 1.14.0	sympy 1.14.0: simplify left (-tan(2x - 3)2 + sec(2x - 3)*2 - 1)tan(2x - 3)/(tan(2x - 3)*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(2x - 3)**2 + 1 = 0
12	✓ 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
qwen3.6:27b-mlx: pass — The solution correctly applies differentiation rules in a step-by-step manner, adhering to the one-change-per-step constraint. The labels used are appropriate for the operations performed, and the final simplification is correct.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies differentiation rules in a step-by-step manner, adhering to the one-change-per-step constraint. The labels used are appropriate for the operations performed, and the final simplification is correct.
gpt-oss:20b: pass 2026-09-26
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies differentiation rules step-by-step, adhering to the one-change-per-step constraint. The labels accurately reflect the operations performed, and the final simplification is mathematically sound.
gpt-oss:20b: fail (style) 2026-09-26 — Step 7 applies both the trig rule (derivative of tan) and the chain rule (derivative of the inner 2*x-3), but it is labeled only as "trig". Each step must change only one rule, so this step violates the granularity requirement.

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.