Home›Calculus 1›Chain rule›Problem 2.158

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

Problem 2.158 · hard

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

\[ \frac{d}{d x} \left(\frac{\ln{\left(\tan^{2}{\left(x - 1 \right)} + 1 \right)}}{2} - \ln{\left(\tan{\left(x - 1 \right)} \right)}\right) \]
derivativeStart with the derivative of the function.✓ Proved
\[ = \frac{d}{d x} \frac{\ln{\left(\tan^{2}{\left(x - 1 \right)} + 1 \right)}}{2} - \frac{d}{d x} \ln{\left(\tan{\left(x - 1 \right)} \right)} \]
sumApply the difference rule.✓ Proved
\[ = \frac{\frac{d}{d x} \ln{\left(\tan^{2}{\left(x - 1 \right)} + 1 \right)}}{2} - \frac{d}{d x} \ln{\left(\tan{\left(x - 1 \right)} \right)} \]
constant-multipleFactor out the constant 1/2.✓ Proved
\[ = - \frac{d}{d x} \ln{\left(\tan{\left(x - 1 \right)} \right)} + \frac{\frac{d}{d x} \left(\tan^{2}{\left(x - 1 \right)} + 1\right)}{2 \left(\tan^{2}{\left(x - 1 \right)} + 1\right)} \]
chainApply the chain rule to the first term.✓ Proved
\[ = - \frac{d}{d x} \ln{\left(\tan{\left(x - 1 \right)} \right)} + \frac{\tan{\left(x - 1 \right)} \frac{d}{d x} \tan{\left(x - 1 \right)}}{\tan^{2}{\left(x - 1 \right)} + 1} \]
chainApply the chain rule to the inner term.✓ Proved
\[ = - \frac{d}{d x} \ln{\left(\tan{\left(x - 1 \right)} \right)} + \frac{\tan{\left(x - 1 \right)} \sec^{2}{\left(x - 1 \right)}}{\tan^{2}{\left(x - 1 \right)} + 1} \]
trigDifferentiate the tangent function.≈ Checked numerically
\[ = - \frac{\frac{d}{d x} \tan{\left(x - 1 \right)}}{\tan{\left(x - 1 \right)}} + \frac{\tan{\left(x - 1 \right)} \sec^{2}{\left(x - 1 \right)}}{\tan^{2}{\left(x - 1 \right)} + 1} \]
chainApply the chain rule to the second term.✓ Proved
\[ = - \frac{\sec^{2}{\left(x - 1 \right)}}{\tan{\left(x - 1 \right)}} + \frac{\tan{\left(x - 1 \right)} \sec^{2}{\left(x - 1 \right)}}{\tan^{2}{\left(x - 1 \right)} + 1} \]
trig algebraDifferentiate the tangent function in the second term. Simplify the multiplication and division.≈ Checked numerically
\[ = \left(- \frac{1}{\tan{\left(x - 1 \right)}} + \frac{\tan{\left(x - 1 \right)}}{\tan^{2}{\left(x - 1 \right)} + 1}\right) \sec^{2}{\left(x - 1 \right)} \]
algebraFactor out sec(x - 1)**2.✓ Proved
\[ = \left(- \cot{\left(x - 1 \right)} + \frac{\tan{\left(x - 1 \right)}}{\tan^{2}{\left(x - 1 \right)} + 1}\right) \sec^{2}{\left(x - 1 \right)} \]
rewriteRewrite 1/tan(x-1) as cot(x-1).✓ Proved
\[ = \left(- \frac{1}{\tan{\left(x - 1 \right)}} + \frac{\tan{\left(x - 1 \right)}}{\tan^{2}{\left(x - 1 \right)} + 1}\right) \sec^{2}{\left(x - 1 \right)} \]
rewriteRevert to fraction for easier algebra.✓ Proved
\[ = - \frac{\sec^{2}{\left(x - 1 \right)}}{\left(\tan^{2}{\left(x - 1 \right)} + 1\right) \tan{\left(x - 1 \right)}} \]
algebra algebra algebra algebraFind a common denominator. Distribute the denominator. Simplify the numerator. Distribute the negative sign.✓ Proved
\[ = - \frac{\sec^{2}{\left(x - 1 \right)}}{\tan^{3}{\left(x - 1 \right)} + \tan{\left(x - 1 \right)}} \]
simplifyExpand the denominator.✓ Proved

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

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: 15 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 - 1)**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 - 1)**2 + 1 = 0
6	≈ 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 log is undefined for non-positive arguments 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
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 sec has poles at odd multiples of pi/2 undefined where tan(x - 1)**2 + 1 = 0 undefined where tan(x - 1) = 0
8	≈ Checked numerically	sympy 1.14.0	sympy 1.14.0: simplify left (-tan(x - 1)2 + sec(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 sec has poles at odd multiples of pi/2 undefined where tan(x - 1) = 0 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) = 0 undefined where tan(x - 1)**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(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 cot has poles at multiples of pi
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 cot has poles at multiples of pi 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) = 0 undefined where tan(x - 1)**2 + 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
16	✓ 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
17	✓ 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 tan(x - 1)3 + tan(x - 1) = 0
answer	≈ Checked numerically	sympy 1.14.0	sympy 1.14.0: final line against the stated answer: simplify left (tan(x - 1)2 - sec(x - 1)2 + 1)/(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
answer, a second way	✓ Proved	sympy 1.14.0	SymPy differentiated f directly and got the stated answer

Reviewers

gpt-oss:20b: fail (error) — The final derivative is incorrectly simplified to “-1/tan(x-1)”. The correct result after the algebraic steps is “-…sec(x-1)^2/(tan(x-1)^3+tan(x-1))”, which does not simplify to “-1/tan(x-1)”. The solution therefore contains a mathematical error.
deepseek-r1:70b: pass
qwen3.6:27b-mlx: pass — The solution correctly applies differentiation rules and algebraic simplifications. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary. The final result matches the stated answer.

Every verdict on record (15)

qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies differentiation rules and algebraic simplifications. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary. The final result matches the stated answer.
gpt-oss:20b: fail (error) 2026-09-20 — The final derivative is incorrectly simplified to “-1/tan(x-1)”. The correct result after the algebraic steps is “-…sec(x-1)^2/(tan(x-1)^3+tan(x-1))”, which does not simplify to “-1/tan(x-1)”. The solution therefore contains a mathematical error.
qwen3.6:27b-mlx: pass 2026-09-20
gpt-oss:20b: pass 2026-09-20
qwen3.6:27b-mlx: fail (style) 2026-09-19 — Step 1 is labeled 'derivative' but performs no differentiation; it merely restates the problem, which should be labeled 'rewrite' or omitted. Step 12 is labeled 'rewrite' but performs an algebraic simplification (reverting cot to 1/tan), which should be labeled 'algebra'.
deepseek-r1:70b: pass 2026-09-19
gpt-oss:20b: fail (error) 2026-09-19 — Several steps apply more than one rule at once (e.g., step 5 combines the power rule with the chain rule, step 7 combines the derivative of log with the chain rule, step 13 combines finding a common denominator with algebraic simplification). Each step should modify only a single term using a single named rule from the allowed vocabulary.
qwen3.6:27b-mlx: fail (error) 2026-09-19 — The final answer is mathematically incorrect. The derivative simplifies to -1/(tan(x-1)*(tan(x-1)^2+1)), but the stated answer is -1/tan(x-1), missing the denominator factor (tan(x-1)^2+1). The solution steps correctly derive the complex form but fail to simplify it to the stated answer, or the stated answer is simply wrong.
deepseek-r1:70b: pass 2026-09-19
gpt-oss:20b: pass 2026-09-19
qwen3.6:27b-mlx: fail (error) 2026-09-18 — The final answer is mathematically incorrect. The derivative of log(tan(x-1)^2 + 1)/2 is 1/tan(x-1), which cancels with the derivative of -log(tan(x-1)), resulting in a total derivative of 0, not -1/tan(x-1). The error occurs in the algebraic simplification steps (specifically step 13-15) where the numerator tan(x-1)^2 - (tan(x-1)^2 + 1) is incorrectly handled or the initial differentiation logic is flawed, leading to a non-zero result for a constant function.
deepseek-r1:70b: pass 2026-09-18
gpt-oss:20b: pass 2026-09-18
gpt-oss:20b: fail 2026-09-17 — The final answer -1/tan(x-1) does not match the derived derivative -sec^2/(tan*(tan^2+1)).
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.