Derivative of \( \displaystyle \frac{3 \operatorname{atan}{\left(4 x + 1 \right)}}{4} \)
Problem 2.1670 · medium
Differentiate \( \displaystyle f(x) = \frac{3 \operatorname{atan}{\left(4 x + 1 \right)}}{4} \).
- \[ \frac{d}{d x} \frac{3 \operatorname{atan}{\left(4 x + 1 \right)}}{4} \]constantStart with the derivative of the function.✓ Proved
- \[ = \frac{3 \frac{d}{d x} \operatorname{atan}{\left(4 x + 1 \right)}}{4} \]constant-multiplePull out the constant factor.✓ Proved
- \[ = \frac{3 \frac{d}{d x} \left(4 x + 1\right)}{4 \left(\left(4 x + 1\right)^{2} + 1\right)} \]inverse-trigApply the derivative rule for arctan(u).✓ Proved
- \[ = \frac{3}{\left(4 x + 1\right)^{2} + 1} \]derivative algebra simplifyDifferentiate the inner function 4x + 1. Cancel the 4 in the numerator with the 4 in the denominator. Combine the terms into a single fraction.✓ Proved
- \[ = \frac{3}{16 x^{2} + 8 x + 2} \]algebra simplifyExpand the squared binomial. Combine like terms.✓ Proved
Answer \( \frac{3}{\left(4 x + 1\right)^{2} + 1} \)
✓ 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 (4*x + 1)**2 + 1 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (4*x + 1)**2 + 1 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (4*x + 1)**2 + 1 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (4*x + 1)**2 + 1 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (4*x + 1)**2 + 1 = 0 undefined where 16*x**2 + 8*x + 2 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 16*x**2 + 8*x + 2 = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where (4*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 (via inverse-trig and derivative steps), and algebraic simplification. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly applies the constant multiple rule, chain rule (via inverse-trig and derivative steps), and algebraic simplification. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: fail (style) 2026-10-06 — Step 1 incorrectly labels the operation as "constant"; the derivative of the whole function is taken, not a constant factor. The correct label would be "derivative" (or "constant-multiple" if the constant were pulled out first).qwen3.6:27b-mlx: pass 2026-10-06
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 2026-10-06 with SymPy 1.14.0.