Derivative of \( \displaystyle \frac{5 \ln{\left(\tan^{2}{\left(4 x - 1 \right)} + 1 \right)}}{8} \)
Problem 2.170 · hard
Differentiate \( \displaystyle f(x) = \frac{5 \ln{\left(\tan^{2}{\left(4 x - 1 \right)} + 1 \right)}}{8} \).
- \[ \frac{d}{d x} \frac{5 \ln{\left(\tan^{2}{\left(4 x - 1 \right)} + 1 \right)}}{8} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{5 \frac{d}{d x} \ln{\left(\tan^{2}{\left(4 x - 1 \right)} + 1 \right)}}{8} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = \frac{5 \frac{d}{d x} \left(\tan^{2}{\left(4 x - 1 \right)} + 1\right)}{8 \left(\tan^{2}{\left(4 x - 1 \right)} + 1\right)} \]chainApply the chain rule for the logarithm.✓ Proved
- \[ = \frac{5 \left(\frac{d}{d x} 1 + \frac{d}{d x} \tan^{2}{\left(4 x - 1 \right)}\right)}{8 \left(\tan^{2}{\left(4 x - 1 \right)} + 1\right)} \]sumDifferentiate the sum inside the parentheses.✓ Proved
- \[ = \frac{5 \frac{d}{d x} \tan^{2}{\left(4 x - 1 \right)}}{8 \left(\tan^{2}{\left(4 x - 1 \right)} + 1\right)} \]constant algebraThe derivative of a constant is zero. Simplify the expression by removing the zero term.✓ Proved
- \[ = \frac{5 \tan{\left(4 x - 1 \right)} \frac{d}{d x} \tan{\left(4 x - 1 \right)}}{4 \left(\tan^{2}{\left(4 x - 1 \right)} + 1\right)} \]powerApply the power rule to the squared tangent term.✓ Proved
- \[ = \frac{5 \tan{\left(4 x - 1 \right)} \sec^{2}{\left(4 x - 1 \right)} \frac{d}{d x} \left(4 x - 1\right)}{4 \left(\tan^{2}{\left(4 x - 1 \right)} + 1\right)} \]chainApply the chain rule to the tangent function.≈ Checked numerically
- \[ = \frac{5 \tan{\left(4 x - 1 \right)} \sec^{2}{\left(4 x - 1 \right)}}{\tan^{2}{\left(4 x - 1 \right)} + 1} \]derivative algebraDifferentiate the inner linear function. Multiply the constants together.✓ Proved
- \[ = \frac{40 \tan{\left(4 x - 1 \right)} \sec^{2}{\left(4 x - 1 \right)}}{8 \tan^{2}{\left(4 x - 1 \right)} + 8} \]algebraCombine the fractions.✓ Proved
- \[ = \frac{5 \tan{\left(4 x - 1 \right)} \sec^{2}{\left(4 x - 1 \right)}}{\tan^{2}{\left(4 x - 1 \right)} + 1} \]simplifyCancel the common factor of 8.✓ Proved
- \[ = \frac{5 \tan{\left(4 x - 1 \right)}}{\left(\tan^{2}{\left(4 x - 1 \right)} + 1\right) \cos^{2}{\left(4 x - 1 \right)}} \]rewriteRewrite secant in terms of cosine.✓ Proved
- \[ = \frac{5 \sin{\left(4 x - 1 \right)}}{\left(\tan^{2}{\left(4 x - 1 \right)} + 1\right) \cos^{3}{\left(4 x - 1 \right)}} \]rewrite algebraRewrite tangent in terms of sine and cosine. Multiply the terms in the numerator and denominator.✓ Proved
- \[ = \frac{5 \sin{\left(4 x - 1 \right)}}{\left(\frac{\sin^{2}{\left(4 x - 1 \right)}}{\cos^{2}{\left(4 x - 1 \right)}} + 1\right) \cos^{3}{\left(4 x - 1 \right)}} \]rewriteRewrite the tangent squared term.✓ Proved
- \[ = \frac{5 \sin{\left(4 x - 1 \right)}}{\left(\sin^{2}{\left(4 x - 1 \right)} + \cos^{2}{\left(4 x - 1 \right)}\right) \cos{\left(4 x - 1 \right)}} \]algebraFind a common denominator for the expression in the denominator.✓ Proved
- \[ = \frac{5 \sin{\left(4 x - 1 \right)}}{\cos{\left(4 x - 1 \right)}} \]simplify algebraUse the Pythagorean identity sin^2 + cos^2 = 1. Simplify the power of cosine in the denominator.✓ Proved
- \[ = 5 \tan{\left(4 x - 1 \right)} \]simplifyRecognize the tangent identity.✓ Proved
Answer \( 5 \tan{\left(4 x - 1 \right)} \)
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: 20 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 undefined where tan(4*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(4*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(4*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(4*x - 1)**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(4*x - 1)**2 + 1 = 0 |
| 8 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left 5*(tan(4*x - 1)**2 - sec(4*x - 1)**2 + 1)*tan(4*x - 1)/(tan(4*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(4*x - 1)**2 + 1 = 0 sec has poles at odd multiples of pi/2 |
| 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(4*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(4*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(4*x - 1)**2 + 1 = 0 undefined where 8*tan(4*x - 1)**2 + 8 = 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 8*tan(4*x - 1)**2 + 8 = 0 undefined where tan(4*x - 1)**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(4*x - 1)**2 + 1 = 0 undefined where cos(4*x - 1) = 0 |
| 14 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 undefined where tan(4*x - 1)**2 + 1 = 0 undefined where cos(4*x - 1) = 0 |
| 15 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 undefined where tan(4*x - 1)**2 + 1 = 0 undefined where cos(4*x - 1) = 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(4*x - 1)**2 + 1 = 0 undefined where cos(4*x - 1) = 0 undefined where sin(4*x - 1)**2/cos(4*x - 1)**2 + 1 = 0 |
| 17 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(4*x - 1) = 0 undefined where sin(4*x - 1)**2/cos(4*x - 1)**2 + 1 = 0 undefined where sin(4*x - 1)**2 + cos(4*x - 1)**2 = 0 |
| 18 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(4*x - 1)**2 + cos(4*x - 1)**2 = 0 undefined where cos(4*x - 1) = 0 |
| 19 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(4*x - 1) = 0 |
| 20 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(4*x - 1) = 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 11 incorrectly rewrites the fraction, applying two algebraic manipulations at once and producing an expression that is not equivalent to the previous step. The correct simplification after step 10 is already 5*tan(4*x-1)*sec(4*x-1)**2/(tan(4*x-1)**2+1).deepseek-r1:70b: passqwen3.6:27b-mlx: pass
Every verdict on record (15)
qwen3.6:27b-mlx: pass 2026-09-20gpt-oss:20b: fail (error) 2026-09-20 — Step 11 incorrectly rewrites the fraction, applying two algebraic manipulations at once and producing an expression that is not equivalent to the previous step. The correct simplification after step 10 is already 5*tan(4*x-1)*sec(4*x-1)**2/(tan(4*x-1)**2+1).qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies differentiation rules and algebraic simplifications in a step-by-step manner, adhering to the one-change-per-step constraint. The final result is correct.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-19 — The solution correctly applies differentiation rules and algebraic simplifications step-by-step. Each step isolates a single rule application, and the labels accurately reflect the operations performed.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: pass 2026-09-19 — The solution correctly applies differentiation rules and algebraic simplifications. Each step adheres to the single-change constraint and uses valid labels from the fixed vocabulary.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: pass 2026-09-18 — The solution correctly applies differentiation rules and algebraic simplifications in single-step increments. The labels accurately reflect the operations performed, and the final result is correct.deepseek-r1:70b: pass 2026-09-18gpt-oss:20b: pass 2026-09-18gpt-oss:20b: pass 2026-09-17deepseek-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.