Home›Calculus 1›Derivatives of trigonometric functions›Problem 2.868

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

Problem 2.868 · hard Beautiful

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

\[ \frac{d}{d x} \left(- \frac{\ln{\left(\tan^{2}{\left(x - 1 \right)} + 1 \right)}}{2}\right) \]
constant-multiplePull out the constant factor.✓ Proved
\[ = - \frac{\frac{d}{d x} \ln{\left(\tan^{2}{\left(x - 1 \right)} + 1 \right)}}{2} \]
logarithmicApply the derivative rule for the logarithm.✓ Proved
\[ = - \frac{\frac{d}{d x} \left(\tan^{2}{\left(x - 1 \right)} + 1\right)}{2 \left(\tan^{2}{\left(x - 1 \right)} + 1\right)} \]
sumDifferentiate the sum inside the parentheses.✓ Proved
\[ = - \frac{\frac{d}{d x} 1 + \frac{d}{d x} \tan^{2}{\left(x - 1 \right)}}{2 \left(\tan^{2}{\left(x - 1 \right)} + 1\right)} \]
constantThe derivative of the constant 1 is 0.✓ Proved
\[ = - \frac{\frac{d}{d x} \tan^{2}{\left(x - 1 \right)}}{2 \left(\tan^{2}{\left(x - 1 \right)} + 1\right)} \]
powerApply the power rule to the squared term.✓ Proved
\[ = - \frac{\tan{\left(x - 1 \right)} \frac{d}{d x} \tan{\left(x - 1 \right)}}{\tan^{2}{\left(x - 1 \right)} + 1} \]
trigApply the chain rule to the tangent function.✓ Proved
\[ = - \frac{\tan{\left(x - 1 \right)} \sec^{2}{\left(x - 1 \right)} \frac{d}{d x} \left(x - 1\right)}{\tan^{2}{\left(x - 1 \right)} + 1} \]
derivativeDifferentiate the inner function (x - 1).≈ Checked numerically
\[ = - \frac{\tan{\left(x - 1 \right)} \sec^{2}{\left(x - 1 \right)}}{\tan^{2}{\left(x - 1 \right)} + 1} \]
algebra simplifySimplify the derivative of (x - 1). Multiply the constants and simplify the expression.✓ Proved
\[ = - \tan{\left(x - 1 \right)} \]
simplify simplifyUse the identity 1 + tan(u)**2 = sec(u)**2. Cancel the common sec(x - 1)**2 term.≈ Checked numerically

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

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: 10 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 undefined where tan(x - 1)**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 - 1)**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 - 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
7	≈ Checked numerically	sympy 1.14.0	sympy 1.14.0: simplify left (-tan(x - 1)2 + sec(x - 1)2 - 1)tan(x - 1)/(tan(x - 1)2 + 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 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 - 1)**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 - 1)**2 + 1 = 0
10	≈ Checked numerically	sympy 1.14.0	sympy 1.14.0: simplify left (tan(x - 1)2 - sec(x - 1)2 + 1)tan(x - 1)/(tan(x - 1)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 - 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
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: fail (error) — Step 3 incorrectly labels the logarithmic derivative as "sum"; step 5 applies both the power and chain rules but labels only "power"; step 6 applies the chain rule to the tangent but labels it "trig". Each step must change only one rule, and the label must match the rule actually applied.
qwen3.6:27b-mlx: pass

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-09-26
gpt-oss:20b: fail (error) 2026-09-26 — Step 3 incorrectly labels the logarithmic derivative as "sum"; step 5 applies both the power and chain rules but labels only "power"; step 6 applies the chain rule to the tangent but labels it "trig". Each step must change only one rule, and the label must match the rule actually applied.
qwen3.6:27b-mlx: pass 2026-09-26
gpt-oss:20b: pass 2026-09-26

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.