Derivative of \( \displaystyle \frac{5 \ln{\left(\tan{\left(3 x + 1 \right)} + \sec{\left(3 x + 1 \right)} \right)}}{3} \)
Problem 2.1233 · hard
Differentiate \( \displaystyle f(x) = \frac{5 \ln{\left(\tan{\left(3 x + 1 \right)} + \sec{\left(3 x + 1 \right)} \right)}}{3} \).
- \[ \frac{d}{d x} \frac{5 \ln{\left(\tan{\left(3 x + 1 \right)} + \sec{\left(3 x + 1 \right)} \right)}}{3} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{5 \frac{d}{d x} \ln{\left(\tan{\left(3 x + 1 \right)} + \sec{\left(3 x + 1 \right)} \right)}}{3} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = \frac{5 \frac{d}{d x} \left(\tan{\left(3 x + 1 \right)} + \sec{\left(3 x + 1 \right)}\right)}{3 \left(\tan{\left(3 x + 1 \right)} + \sec{\left(3 x + 1 \right)}\right)} \]logarithmicApply the chain rule for the natural logarithm.✓ Proved
- \[ = \frac{5 \left(\frac{d}{d x} \tan{\left(3 x + 1 \right)} + \frac{d}{d x} \sec{\left(3 x + 1 \right)}\right)}{3 \left(\tan{\left(3 x + 1 \right)} + \sec{\left(3 x + 1 \right)}\right)} \]sumApply the sum rule to the derivative.✓ Proved
- \[ = \frac{5 \left(3 \tan{\left(3 x + 1 \right)} \sec{\left(3 x + 1 \right)} + 3 \sec^{2}{\left(3 x + 1 \right)}\right)}{3 \left(\tan{\left(3 x + 1 \right)} + \sec{\left(3 x + 1 \right)}\right)} \]chainApply the chain rule to both tan and sec terms.≈ Checked numerically
- \[ = \frac{5 \left(\tan{\left(3 x + 1 \right)} \sec{\left(3 x + 1 \right)} + \sec^{2}{\left(3 x + 1 \right)}\right)}{\tan{\left(3 x + 1 \right)} + \sec{\left(3 x + 1 \right)}} \]constant-multipleFactor out the common constant 3.✓ Proved
- \[ = \frac{5 \tan{\left(3 x + 1 \right)} \sec{\left(3 x + 1 \right)} + 5 \sec^{2}{\left(3 x + 1 \right)}}{\tan{\left(3 x + 1 \right)} + \sec{\left(3 x + 1 \right)}} \]algebraSimplify the expression by multiplying the constants.✓ Proved
- \[ = 5 \sec{\left(3 x + 1 \right)} \]algebra simplifyFactor out sec(3*x + 1) from the numerator. Cancel the common term in the numerator and denominator.✓ Proved
Answer \( \frac{5}{\cos{\left(3 x + 1 \right)}} \)
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: 9 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(3*x + 1) + sec(3*x + 1) = 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(3*x + 1) + sec(3*x + 1) = 0 |
| 5 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left 5*(tan(3*x + 1)**2 - sec(3*x + 1)**2 + 1)/(tan(3*x + 1) + sec(3*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(3*x + 1) + sec(3*x + 1) = 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(3*x + 1) + sec(3*x + 1) = 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(3*x + 1) + sec(3*x + 1) = 0 |
| 8 | ✓ 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(3*x + 1) + sec(3*x + 1) = 0 |
| 9 | ✓ 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(3*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: passqwen3.6:27b-mlx: pass — The solution correctly applies differentiation rules and algebraic simplifications. Each step changes only one aspect of the expression, and the labels accurately reflect the operations performed.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies differentiation rules and algebraic simplifications. Each step changes only one aspect of the expression, and the labels accurately reflect the operations performed.gpt-oss:20b: pass 2026-09-29qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies differentiation rules and algebraic simplifications in single-step increments. The labels accurately reflect the operations performed at each stage.gpt-oss:20b: fail (error) 2026-09-29 — Step 5 applies the chain rule to both the tan and sec terms in a single line, combining two separate rule applications into one step. This violates the rule that each step must change only one thing and apply only one named 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 2026-09-29 with SymPy 1.14.0.