Derivative of \( \displaystyle \frac{5 \operatorname{atan}{\left(3 x + 1 \right)}}{3} \)
Problem 2.677 · medium
Differentiate \( \displaystyle f(x) = \frac{5 \operatorname{atan}{\left(3 x + 1 \right)}}{3} \).
- \[ \frac{d}{d x} \frac{5 \operatorname{atan}{\left(3 x + 1 \right)}}{3} \]Start with the derivative of the function.✓ Proved
- \[ = \frac{5 \frac{d}{d x} \operatorname{atan}{\left(3 x + 1 \right)}}{3} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = \frac{5 \frac{d}{d x} \left(3 x + 1\right)}{3 \left(\left(3 x + 1\right)^{2} + 1\right)} \]chainApply the chain rule to the arctangent function.✓ Proved
- \[ = \frac{5}{\left(3 x + 1\right)^{2} + 1} \]derivative algebraDifferentiate the inner function 3*x + 1. Simplify the expression by canceling the 3.✓ Proved
- \[ = \frac{5}{9 x^{2} + 6 x + 2} \]algebra simplifyExpand the squared term. Combine like terms in the denominator.✓ Proved
Answer \( \frac{5}{\left(3 x + 1\right)^{2} + 1} \)
✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. 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 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (3*x + 1)**2 + 1 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (3*x + 1)**2 + 1 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (3*x + 1)**2 + 1 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (3*x + 1)**2 + 1 = 0 undefined where 9*x**2 + 6*x + 2 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 9*x**2 + 6*x + 2 = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where (3*x + 1)**2 + 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 the constant multiple rule, chain rule, and basic differentiation rules. The algebraic simplifications are valid and clearly labeled.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-21 — The solution correctly applies the constant multiple rule, chain rule, and basic differentiation rules. The algebraic simplifications are valid and clearly labeled.gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: pass 2026-09-21 — The solution correctly applies the constant multiple rule, chain rule, and derivative of arctangent. Each step isolates a single operation, and the labels are accurate.gpt-oss:20b: pass 2026-09-21
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.