Home›Calculus 1›Derivatives of trigonometric functions›Problem 2.1221

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

Problem 2.1221 · hard Beautiful

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

\[ \frac{d}{d x} \left(- \frac{\ln{\left(\tan^{2}{\left(x - 3 \right)} + 1 \right)}}{2}\right) \]
constant-multiple✓ Proved
\[ = - \frac{\frac{d}{d x} \ln{\left(\tan^{2}{\left(x - 3 \right)} + 1 \right)}}{2} \]
chainApply the chain rule to the logarithm.✓ Proved
\[ = - \frac{\frac{d}{d x} \left(\tan^{2}{\left(x - 3 \right)} + 1\right)}{2 \left(\tan^{2}{\left(x - 3 \right)} + 1\right)} \]
sumDifferentiate the sum inside the derivative.✓ Proved
\[ = - \frac{\frac{d}{d x} 1 + \frac{d}{d x} \tan^{2}{\left(x - 3 \right)}}{2 \left(\tan^{2}{\left(x - 3 \right)} + 1\right)} \]
constantThe derivative of a constant is zero.✓ Proved
\[ = - \frac{\frac{d}{d x} \tan^{2}{\left(x - 3 \right)}}{2 \left(\tan^{2}{\left(x - 3 \right)} + 1\right)} \]
powerApply the power rule.✓ Proved
\[ = - \frac{\tan{\left(x - 3 \right)} \frac{d}{d x} \tan{\left(x - 3 \right)}}{\tan^{2}{\left(x - 3 \right)} + 1} \]
trigDifferentiate the tangent function.✓ Proved
\[ = - \frac{\tan{\left(x - 3 \right)} \sec^{2}{\left(x - 3 \right)} \frac{d}{d x} \left(x - 3\right)}{\tan^{2}{\left(x - 3 \right)} + 1} \]
derivativeDifferentiate the inner linear function.≈ Checked numerically
\[ = - \frac{\tan{\left(x - 3 \right)} \sec^{2}{\left(x - 3 \right)}}{\tan^{2}{\left(x - 3 \right)} + 1} \]
algebra algebraSimplify the derivative of x - 3. Simplify the coefficients.✓ Proved
\[ = - \tan{\left(x - 3 \right)} \]
simplify simplifyUse the identity tan(u)**2 + 1 = sec(u)**2. Cancel the common sec(x - 3)**2 term.≈ Checked numerically

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

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.

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 - 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(x - 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(x - 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(x - 3)**2 + 1 = 0
7	≈ 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 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
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 - 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)**2 + 1 = 0
10	≈ 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 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
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: pass
qwen3.6:27b-mlx: fail (style) — Step 2 is labeled 'chain' but performs the constant-multiple rule (factoring out -1/2); Step 3 is labeled 'sum' but performs the chain rule on the logarithm. The labels do not match the operations performed.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (style) 2026-09-29 — Step 2 is labeled 'chain' but performs the constant-multiple rule (factoring out -1/2); Step 3 is labeled 'sum' but performs the chain rule on the logarithm. The labels do not match the operations performed.
gpt-oss:20b: pass 2026-09-29
qwen3.6:27b-mlx: fail (error) 2026-09-29 — Step 1 is labeled 'constant-multiple' but performs no transformation; the expression is identical to the input. Step 2 is labeled 'chain' but only factors out the constant -1/2, which is the application of the constant-multiple rule, not the chain rule.
gpt-oss:20b: fail (style) 2026-09-29 — Step 2 incorrectly labels the constant‑multiple step as "chain", and step 3 incorrectly labels the derivative of the logarithm as "sum". The correct labels should be "constant‑multiple" for step 2 and "chain" (or "derivative" for the log) for step 3.

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.