Derivative of \( \displaystyle \left(3 x + 2\right) \tan{\left(3 x + 2 \right)} \)
Problem 2.511 · medium
Differentiate \( \displaystyle f(x) = \left(3 x + 2\right) \tan{\left(3 x + 2 \right)} \).
- \[ \frac{d}{d x} \left(3 x + 2\right) \tan{\left(3 x + 2 \right)} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \left(3 x + 2\right) \frac{d}{d x} \tan{\left(3 x + 2 \right)} + \tan{\left(3 x + 2 \right)} \frac{d}{d x} \left(3 x + 2\right) \]productApply the product rule.✓ Proved
- \[ = \left(3 x + 2\right) \frac{d}{d x} \tan{\left(3 x + 2 \right)} + 3 \tan{\left(3 x + 2 \right)} \]derivativeDifferentiate the first term.✓ Proved
- \[ = \left(3 x + 2\right) \sec^{2}{\left(3 x + 2 \right)} \frac{d}{d x} \left(3 x + 2\right) + 3 \tan{\left(3 x + 2 \right)} \]chainApply the chain rule to the tangent term.≈ Checked numerically
- \[ = 3 \left(3 x + 2\right) \sec^{2}{\left(3 x + 2 \right)} + 3 \tan{\left(3 x + 2 \right)} \]derivativeDifferentiate the inner function of the chain rule.✓ Proved
- \[ = \left(9 x + 6\right) \sec^{2}{\left(3 x + 2 \right)} + 3 \tan{\left(3 x + 2 \right)} \]algebra algebraRearrange the terms. Distribute the constant 3.✓ Proved
Answer \( \frac{3 \left(3 x + \frac{\sin{\left(6 x + 4 \right)}}{2} + 2\right)}{\cos^{2}{\left(3 x + 2 \right)}} \)
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.
✓ Nihil obstat Lines: 6 proved, 2 checked numerically. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
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 tan has poles at odd multiples of pi/2 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 |
| 4 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left 3*(3*x + 2)*(tan(3*x + 2)**2 - sec(3*x + 2)**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 |
| 5 | ✓ 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 |
| 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 |
| 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 |
| answer | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: final line against the stated answer: simplify left 3*(-6*x + 2*((3*x + 2)*sec(3*x + 2)**2 + tan(3*x + 2))*cos(3*x + 2)**2 - sin(6*x + 4) - 4)/(2*cos(3*x + 2)**2); numeric agreement only, at 24 of 24 sampled points undefined where cos(3*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: pass — The solution correctly applies the product rule, chain rule, and basic differentiation rules in a step-by-step manner. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-21 — The solution correctly applies the product rule, chain rule, and basic differentiation rules in a step-by-step manner. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.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 the product rule and chain rule in separate steps. The final answer is algebraically equivalent to the stated answer, and all steps adhere to the single-rule constraint.gpt-oss:20b: pass 2026-09-20
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.