Derivative of \( \displaystyle \frac{\ln{\left(\tan{\left(4 x + 2 \right)} + \sec{\left(4 x + 2 \right)} \right)}}{2} \)
Problem 2.523 · hard
Differentiate \( \displaystyle f(x) = \frac{\ln{\left(\tan{\left(4 x + 2 \right)} + \sec{\left(4 x + 2 \right)} \right)}}{2} \).
- \[ \frac{d}{d x} \frac{\ln{\left(\tan{\left(4 x + 2 \right)} + \sec{\left(4 x + 2 \right)} \right)}}{2} \]constant-multipleMove the constant factor outside the derivative.✓ Proved
- \[ = \frac{\frac{d}{d x} \ln{\left(\tan{\left(4 x + 2 \right)} + \sec{\left(4 x + 2 \right)} \right)}}{2} \]chainApply the chain rule to the natural logarithm.✓ Proved
- \[ = \frac{\frac{d}{d x} \left(\tan{\left(4 x + 2 \right)} + \sec{\left(4 x + 2 \right)}\right)}{2 \left(\tan{\left(4 x + 2 \right)} + \sec{\left(4 x + 2 \right)}\right)} \]sumDifferentiate the sum inside the numerator.✓ Proved
- \[ = \frac{\frac{\frac{d}{d x} \tan{\left(4 x + 2 \right)}}{2} + \frac{\frac{d}{d x} \sec{\left(4 x + 2 \right)}}{2}}{\tan{\left(4 x + 2 \right)} + \sec{\left(4 x + 2 \right)}} \]chainApply the chain rule to both the tangent and secant terms.✓ Proved
- \[ = \frac{2 \tan{\left(4 x + 2 \right)} \sec{\left(4 x + 2 \right)} + 2 \sec^{2}{\left(4 x + 2 \right)}}{\tan{\left(4 x + 2 \right)} + \sec{\left(4 x + 2 \right)}} \]algebra algebraFactor out the common constant 4 from the numerator. Factor out the secant term in the numerator.≈ Checked numerically
- \[ = 2 \sec{\left(4 x + 2 \right)} \]simplify simplifyCancel the common factor in the numerator and denominator. Final simplified result.✓ Proved
Answer \( \frac{2}{\cos{\left(4 x + 2 \right)}} \)
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: 8 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 sec 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 sec has poles at odd multiples of pi/2 undefined where tan(4*x + 2) + sec(4*x + 2) = 0 |
| 4 | ✓ 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 + 2) + sec(4*x + 2) = 0 |
| 5 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left 2*(tan(4*x + 2)**2 - sec(4*x + 2)**2 + 1)/(tan(4*x + 2) + sec(4*x + 2)); 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(4*x + 2) + sec(4*x + 2) = 0 |
| 6 | ✓ 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 + 2) + sec(4*x + 2) = 0 |
| 7 | ✓ 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 + 2) + sec(4*x + 2) = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 sec 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 undefined where cos(4*x + 2) = 0 |
| answer, a second way | ✓ Proved | sympy 1.14.0 | SymPy differentiated f directly and got the stated answer |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: fail (error) — Step 4 applies the chain rule to differentiate the inner functions, but it leaves the derivative operators `Derivative(..., x)` unevaluated. The label 'chain' is insufficient because the step also requires applying the standard derivatives of tan and sec (rule 'derivative') to resolve the operators. A single step cannot combine the structural application of the chain rule with the evaluation of the specific trigonometric derivatives without violating the one-rule-per-step constraint or mislabeling the evaluation part.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-09-21 — Step 4 applies the chain rule to differentiate the inner functions, but it leaves the derivative operators `Derivative(..., x)` unevaluated. The label 'chain' is insufficient because the step also requires applying the standard derivatives of tan and sec (rule 'derivative') to resolve the operators. A single step cannot combine the structural application of the chain rule with the evaluation of the specific trigonometric derivatives without violating the one-rule-per-step constraint or mislabeling the evaluation part.gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: pass 2026-09-20gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies differentiation rules and algebraic simplifications. Each step isolates a single operation, and the labels accurately reflect the rules applied (e.g., chain rule for composite functions, algebra for factoring and canceling).gpt-oss:20b: fail (error) 2026-09-20 — Step 5 claims to factor out the constant 4, but the expression is left unchanged. Step 6 performs two operations—factoring out 4 and simplifying the 1/2 factor—in a single step, violating the one‑rule‑per‑step rule.
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.