Home›Calculus 1›Chain rule›Problem 2.299

Derivative of \( \displaystyle - \frac{5 \ln{\left(\tan^{2}{\left(x + 1 \right)} + 1 \right)}}{2} + 5 \ln{\left(\tan{\left(x + 1 \right)} \right)} \)

Problem 2.299 · hard Beautiful

Differentiate \( \displaystyle f(x) = - \frac{5 \ln{\left(\tan^{2}{\left(x + 1 \right)} + 1 \right)}}{2} + 5 \ln{\left(\tan{\left(x + 1 \right)} \right)} \).

\[ \frac{d}{d x} \left(- \frac{5 \ln{\left(\tan^{2}{\left(x + 1 \right)} + 1 \right)}}{2} + 5 \ln{\left(\tan{\left(x + 1 \right)} \right)}\right) \]
constant-multipleStart with the derivative of the function. Distribute the constant factor.✓ Proved
\[ = \frac{d}{d x} \left(- \frac{5 \ln{\left(\tan^{2}{\left(x + 1 \right)} + 1 \right)}}{2}\right) + \frac{d}{d x} 5 \ln{\left(\tan{\left(x + 1 \right)} \right)} \]
sumApply the sum rule for derivatives.✓ Proved
\[ = \frac{d}{d x} \left(- \frac{5 \ln{\left(\tan^{2}{\left(x + 1 \right)} + 1 \right)}}{2}\right) + 5 \frac{d}{d x} \ln{\left(\tan{\left(x + 1 \right)} \right)} \]
constant-multipleFactor out the constant from the second term.✓ Proved
\[ = \frac{d}{d x} \left(- \frac{5 \ln{\left(\tan^{2}{\left(x + 1 \right)} + 1 \right)}}{2}\right) + \frac{5 \frac{d}{d x} \tan{\left(x + 1 \right)}}{\tan{\left(x + 1 \right)}} \]
logarithmicApply the chain rule for the logarithm.✓ Proved
\[ = \frac{d}{d x} \left(- \frac{5 \ln{\left(\tan^{2}{\left(x + 1 \right)} + 1 \right)}}{2}\right) + \frac{5 \sec^{2}{\left(x + 1 \right)} \frac{d}{d x} \left(x + 1\right)}{\tan{\left(x + 1 \right)}} \]
trigApply the chain rule for the tangent function.≈ Checked numerically
\[ = \frac{d}{d x} \left(- \frac{5 \ln{\left(\tan^{2}{\left(x + 1 \right)} + 1 \right)}}{2}\right) + \frac{5 \sec^{2}{\left(x + 1 \right)}}{\tan{\left(x + 1 \right)}} \]
derivative algebraDifferentiate the innermost function x + 1. Simplify the second term.✓ Proved
\[ = \frac{5 \sec^{2}{\left(x + 1 \right)}}{\tan{\left(x + 1 \right)}} - \frac{5 \frac{d}{d x} \left(\tan^{2}{\left(x + 1 \right)} + 1\right)}{2 \left(\tan^{2}{\left(x + 1 \right)} + 1\right)} \]
logarithmicApply the chain rule to the first term's logarithm.✓ Proved
\[ = \frac{5 \sec^{2}{\left(x + 1 \right)}}{\tan{\left(x + 1 \right)}} - \frac{5 \left(\frac{d}{d x} 1 + \frac{d}{d x} \tan^{2}{\left(x + 1 \right)}\right)}{2 \left(\tan^{2}{\left(x + 1 \right)} + 1\right)} \]
sumDifferentiate the inside of the logarithm.✓ Proved
\[ = \frac{5 \sec^{2}{\left(x + 1 \right)}}{\tan{\left(x + 1 \right)}} - \frac{5 \frac{d}{d x} \tan^{2}{\left(x + 1 \right)}}{2 \left(\tan^{2}{\left(x + 1 \right)} + 1\right)} \]
constantThe derivative of 1 is 0.✓ Proved
\[ = \frac{5 \sec^{2}{\left(x + 1 \right)}}{\tan{\left(x + 1 \right)}} - \frac{5 \tan{\left(x + 1 \right)} \frac{d}{d x} \tan{\left(x + 1 \right)}}{\tan^{2}{\left(x + 1 \right)} + 1} \]
powerApply the power rule to tan(x + 1)**2.✓ Proved
\[ = \frac{5 \sec^{2}{\left(x + 1 \right)}}{\tan{\left(x + 1 \right)}} - \frac{5 \tan{\left(x + 1 \right)} \sec^{2}{\left(x + 1 \right)}}{\tan^{2}{\left(x + 1 \right)} + 1} \]
trig derivative algebraApply the chain rule to tan(x + 1). Differentiate x + 1. Simplify the first term by canceling 2.≈ Checked numerically
\[ = 5 \left(\frac{1}{\tan{\left(x + 1 \right)}} - \frac{\tan{\left(x + 1 \right)}}{\tan^{2}{\left(x + 1 \right)} + 1}\right) \sec^{2}{\left(x + 1 \right)} \]
algebraFactor out the common term 5*sec(x+1)**2.✓ Proved
\[ = \frac{5 \sec^{2}{\left(x + 1 \right)}}{\left(\tan^{2}{\left(x + 1 \right)} + 1\right) \tan{\left(x + 1 \right)}} \]
algebra algebra algebra simplifyCombine the fractions inside the parentheses. Expand the numerator. Simplify the numerator to 1. Final simplified form.✓ Proved

Answer \( \frac{5}{\tan{\left(x + 1 \right)}} \)

Lines: 18 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
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) = 0
6	≈ Checked numerically	sympy 1.14.0	sympy 1.14.0: simplify left 5(tan(x + 1)2 - sec(x + 1)*2 + 1)/tan(x + 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) = 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) = 0
8	✓ 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) = 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 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
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 undefined where tan(x + 1) = 0 undefined where tan(x + 1)**2 + 1 = 0
13	≈ Checked numerically	sympy 1.14.0	sympy 1.14.0: simplify left 5(-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) = 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) = 0 undefined where tan(x + 1)**2 + 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) = 0 undefined where tan(x + 1)**2 + 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) = 0 undefined where tan(x + 1)**2 + 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) = 0 undefined where tan(x + 1)**2 + 1 = 0
18	✓ 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
19	✓ 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
20	✓ 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
answer	≈ Checked numerically	sympy 1.14.0	sympy 1.14.0: final line against the stated answer: simplify left (-5tan(x + 1)2 + 5sec(x + 1)2 - 5)/(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: pass
deepseek-r1:70b: pass
qwen3.6:27b-mlx: fail (error) — The final answer is incorrect. The solution fails to simplify the expression using the identity tan(x)^2 + 1 = sec(x)^2, which would cancel the sec(x+1)^2 term, leaving 5/tan(x+1). Instead, it stops at 5*sec(x+1)^2 / (tan(x+1)*(tan(x+1)^2+1)).

Every verdict on record (12)

qwen3.6:27b-mlx: fail (error) 2026-09-20 — The final answer is incorrect. The solution fails to simplify the expression using the identity tan(x)^2 + 1 = sec(x)^2, which would cancel the sec(x+1)^2 term, leaving 5/tan(x+1). Instead, it stops at 5*sec(x+1)^2 / (tan(x+1)*(tan(x+1)^2+1)).
gpt-oss:20b: pass 2026-09-20
qwen3.6:27b-mlx: fail (error) 2026-09-20 — The final answer is mathematically incorrect. The derivative of log(tan(x+1)^2 + 1) is 2*tan(x+1)*sec(x+1)^2 / (tan(x+1)^2 + 1), which simplifies to sin(2(x+1)). The derivative of log(tan(x+1)) is sec(x+1)^2/tan(x+1) = csc(2(x+1)). The sum is 5*(sin(2(x+1)) + csc(2(x+1))), not 5/tan(x+1). The error occurs in the simplification steps (15-20) where the terms are incorrectly combined or simplified to match the stated (wrong) answer.
gpt-oss:20b: pass 2026-09-20
qwen3.6:27b-mlx: fail (error) 2026-09-19 — The final answer is mathematically incorrect. The derivative of the first term simplifies to -5/(2*cos(2x+2)), which does not cancel with the second term to yield 5/tan(x+1). The algebraic simplification in steps 16-20 is flawed because it ignores the factor of 1/2 from the first term's coefficient, leading to an incorrect cancellation.
deepseek-r1:70b: pass 2026-09-19
gpt-oss:20b: pass 2026-09-19
qwen3.6:27b-mlx: fail (error) 2026-09-19 — The final answer is mathematically incorrect. The derivative of the first term is -5*sec(x+1)^2 / (tan(x+1)^2 + 1), which simplifies to -5*cos(x+1)^2. The derivative of the second term is 5*sec(x+1)^2 / tan(x+1). The sum is 5*sec(x+1)^2 * (1/tan(x+1) - cos(x+1)^2), which does not simplify to 5/tan(x+1). The solver incorrectly cancelled the (tan(x+1)^2 + 1) term in the denominator of the first part during simplification steps 15-20, effectively treating tan(x+1)^2 + 1 as 1 or cancelling it against a non-existent factor.
deepseek-r1:70b: pass 2026-09-19
gpt-oss:20b: pass 2026-09-19
gpt-oss:20b: fail 2026-09-17 — The final answer 5/tan(x+1) is incorrect; the correct derivative is 5*sec(x+1)**2/(tan(x+1)*(tan(x+1)**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.