Home›Calculus 1›Derivatives of trigonometric functions›Problem 2.1059

Derivative of \( \displaystyle - \frac{5 \ln{\left(\tan^{2}{\left(x - 3 \right)} + 1 \right)}}{2} + 5 \ln{\left(\tan{\left(x - 3 \right)} \right)} \)

Problem 2.1059 · hard Beautiful

Differentiate \( \displaystyle f(x) = - \frac{5 \ln{\left(\tan^{2}{\left(x - 3 \right)} + 1 \right)}}{2} + 5 \ln{\left(\tan{\left(x - 3 \right)} \right)} \).

\[ \frac{d}{d x} \left(- \frac{5 \ln{\left(\tan^{2}{\left(x - 3 \right)} + 1 \right)}}{2} + 5 \ln{\left(\tan{\left(x - 3 \right)} \right)}\right) \]
sumStart with the derivative of the sum.✓ Proved
\[ = \frac{d}{d x} \left(- \frac{5 \ln{\left(\tan^{2}{\left(x - 3 \right)} + 1 \right)}}{2}\right) + \frac{d}{d x} 5 \ln{\left(\tan{\left(x - 3 \right)} \right)} \]
constant-multipleDistribute the constant factor.✓ Proved
\[ = \frac{d}{d x} \left(- \frac{5 \ln{\left(\tan^{2}{\left(x - 3 \right)} + 1 \right)}}{2}\right) + 5 \frac{d}{d x} \ln{\left(\tan{\left(x - 3 \right)} \right)} \]
constant-multipleFactor out the constant from the second term.✓ Proved
\[ = - \frac{5 \frac{d}{d x} \ln{\left(\tan^{2}{\left(x - 3 \right)} + 1 \right)}}{2} + 5 \frac{d}{d x} \ln{\left(\tan{\left(x - 3 \right)} \right)} \]
constant-multipleFactor out the constant from the first term.✓ Proved
\[ = \frac{5 \frac{d}{d x} \tan{\left(x - 3 \right)}}{\tan{\left(x - 3 \right)}} - \frac{5 \frac{d}{d x} \left(\tan^{2}{\left(x - 3 \right)} + 1\right)}{2 \left(\tan^{2}{\left(x - 3 \right)} + 1\right)} \]
logarithmicApply the derivative rule for logarithms.✓ Proved
\[ = \frac{5 \frac{d}{d x} \tan{\left(x - 3 \right)}}{\tan{\left(x - 3 \right)}} - \frac{5 \frac{d}{d x} \tan^{2}{\left(x - 3 \right)}}{2 \left(\tan^{2}{\left(x - 3 \right)} + 1\right)} \]
sumDifferentiate the sum inside the first logarithm.✓ Proved
\[ = \frac{5 \frac{d}{d x} \tan{\left(x - 3 \right)}}{\tan{\left(x - 3 \right)}} - \frac{5 \tan{\left(x - 3 \right)} \frac{d}{d x} \tan{\left(x - 3 \right)}}{\tan^{2}{\left(x - 3 \right)} + 1} \]
power chainApply the power rule to the squared term. The derivative of x-3 is 1.✓ Proved
\[ = \frac{5 \sec^{2}{\left(x - 3 \right)}}{\tan{\left(x - 3 \right)}} - \frac{5 \tan{\left(x - 3 \right)} \sec^{2}{\left(x - 3 \right)}}{\tan^{2}{\left(x - 3 \right)} + 1} \]
derivative algebraDifferentiate the tangent function. Simplify the fraction by multiplying -5/2 by 2.≈ Checked numerically
\[ = 5 \left(\frac{1}{\tan{\left(x - 3 \right)}} - \frac{\tan{\left(x - 3 \right)}}{\tan^{2}{\left(x - 3 \right)} + 1}\right) \sec^{2}{\left(x - 3 \right)} \]
constant-multipleFactor out 5 * sec(x - 3)**2.✓ Proved
\[ = \frac{5 \sec^{2}{\left(x - 3 \right)}}{\left(\tan^{2}{\left(x - 3 \right)} + 1\right) \tan{\left(x - 3 \right)}} \]
algebra algebra simplifyFind a common denominator for the terms in the parentheses. Expand the numerator. Cancel the tan(x-3)**2 terms.✓ Proved
\[ = \frac{5}{\left(\tan^{2}{\left(x - 3 \right)} + 1\right) \cos^{2}{\left(x - 3 \right)} \tan{\left(x - 3 \right)}} \]
rewrite algebraRewrite sec(x-3)**2 in terms of cosine. Combine the fractions.✓ Proved
\[ = \frac{5}{\left(\tan^{2}{\left(x - 3 \right)} + 1\right) \sin{\left(x - 3 \right)} \cos{\left(x - 3 \right)}} \]
rewrite algebraRewrite tangent in terms of sine and cosine. Simplify the cosine terms in the denominator.✓ Proved
\[ = \frac{5}{\left(\frac{\sin^{2}{\left(x - 3 \right)}}{\cos^{2}{\left(x - 3 \right)}} + 1\right) \sin{\left(x - 3 \right)} \cos{\left(x - 3 \right)}} \]
rewriteRewrite the tangent squared term.✓ Proved
\[ = \frac{5 \cos{\left(x - 3 \right)}}{\left(\sin^{2}{\left(x - 3 \right)} + \cos^{2}{\left(x - 3 \right)}\right) \sin{\left(x - 3 \right)}} \]
algebraCombine terms in the last parenthesis.✓ Proved
\[ = \frac{5 \cos{\left(x - 3 \right)}}{\sin{\left(x - 3 \right)}} \]
trig algebra algebraUse the Pythagorean identity sin^2 + cos^2 = 1. Multiply by the reciprocal of the denominator. Simplify the cosine terms.✓ Proved
\[ = 5 \cot{\left(x - 3 \right)} \]
trigExpress the result in terms of the cotangent function.✓ Proved

Answer \( \frac{5}{\tan{\left(x - 3 \right)}} \)

Lines: 24 proved, 1 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 - 3) = 0 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) = 0 undefined where tan(x - 3)**2 + 1 = 0
7	✓ 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) = 0 undefined where tan(x - 3)**2 + 1 = 0
8	✓ 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) = 0 undefined where tan(x - 3)**2 + 1 = 0
9	≈ Checked numerically	sympy 1.14.0	sympy 1.14.0: simplify left (5tan(x - 3)2 - 5sec(x - 3)2 + 5)/(tan(x - 3)3 + tan(x - 3)); numeric agreement only, at 24 of 24 sampled points tan has poles at odd multiples of pi/2 undefined where tan(x - 3) = 0 undefined where tan(x - 3)**2 + 1 = 0 sec has poles at odd multiples of pi/2
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 - 3) = 0 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 sec has poles at odd multiples of pi/2 undefined where tan(x - 3) = 0 undefined where tan(x - 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 sec has poles at odd multiples of pi/2 undefined where tan(x - 3) = 0 undefined where tan(x - 3)**2 + 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 - 3) = 0 undefined where tan(x - 3)**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 - 3) = 0 undefined where tan(x - 3)**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 - 3) = 0 undefined where tan(x - 3)**2 + 1 = 0 undefined where cos(x - 3) = 0
16	✓ 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) = 0 undefined where tan(x - 3)**2 + 1 = 0 undefined where cos(x - 3) = 0
17	✓ 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) = 0 undefined where tan(x - 3)**2 + 1 = 0 undefined where cos(x - 3) = 0 undefined where sin(x - 3) = 0
18	✓ 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 undefined where cos(x - 3) = 0 undefined where sin(x - 3) = 0
19	✓ 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 undefined where cos(x - 3) = 0 undefined where sin(x - 3) = 0 undefined where sin(x - 3)2/cos(x - 3)**2 + 1 = 0
20	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where sin(x - 3)2/cos(x - 3)2 + 1 = 0 undefined where sin(x - 3) = 0 undefined where cos(x - 3) = 0 undefined where sin(x - 3)2 + cos(x - 3)2 = 0
21	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where sin(x - 3)2 + cos(x - 3)2 = 0 undefined where sin(x - 3) = 0
22	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where sin(x - 3) = 0
23	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where sin(x - 3) = 0
24	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where sin(x - 3) = 0 cot has poles at multiples of pi
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 undefined where tan(x - 3) = 0
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: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The final answer 5*cot(x-3) is mathematically incorrect; the correct derivative is 5/tan(x-3), which simplifies to 5*cot(x-3) only if one ignores th

Every verdict on record (4)

qwen3.6:27b-mlx: inconclusive 2026-09-28 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The final answer 5*cot(x-3) is mathematically incorrect; the correct derivative is 5/tan(x-3), which simplifies to 5*cot(x-3) only if one ignores th
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (style) 2026-09-28 — Step 8 is labeled 'chain' but performs no differentiation; it merely rewrites 'x - 3' as 'x * 1 - 3', which is an algebraic rewrite or simplification, not an application of the chain rule. The label does not match the operation.
gpt-oss:20b: pass 2026-09-28

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 2026-09-28 with SymPy 1.14.0.