Derivative of \( \displaystyle - \frac{\operatorname{atan}{\left(5 x + 2 \right)}}{5} \)
Problem 2.828 · medium Video
Differentiate \( \displaystyle f(x) = - \frac{\operatorname{atan}{\left(5 x + 2 \right)}}{5} \).
- \[ \frac{d}{d x} \left(- \frac{\operatorname{atan}{\left(5 x + 2 \right)}}{5}\right) \]Start with the derivative of the function.✓ Proved
- \[ = - \frac{\frac{d}{d x} \operatorname{atan}{\left(5 x + 2 \right)}}{5} \]constantPull out the constant factor.✓ Proved
- \[ = - \frac{\frac{d}{d x} \left(5 x + 2\right)}{5 \left(\left(5 x + 2\right)^{2} + 1\right)} \]inverse-trigApply the derivative rule for arctan.✓ Proved
- \[ = - \frac{1}{\left(5 x + 2\right)^{2} + 1} \]derivative algebraDifferentiate the inner function 5*x + 2. Simplify the expression by canceling the 5.✓ Proved
- \[ = - \frac{1}{25 x^{2} + 20 x + 5} \]algebra simplifyExpand the squared binomial. Combine like terms in the denominator.✓ Proved
Answer \( - \frac{1}{\left(5 x + 2\right)^{2} + 1} \)
✓ 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. How we verify
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 (5*x + 2)**2 + 1 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (5*x + 2)**2 + 1 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (5*x + 2)**2 + 1 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (5*x + 2)**2 + 1 = 0 undefined where 25*x**2 + 20*x + 5 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 25*x**2 + 20*x + 5 = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where (5*x + 2)**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, the chain rule for arctan, and basic algebraic simplifications. 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 2026-09-27 — The solution correctly applies the constant multiple rule, the chain rule for arctan, and basic algebraic simplifications. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: pass 2026-09-26gpt-oss:20b: pass 2026-09-26qwen3.6:27b-mlx: pass 2026-09-26gpt-oss:20b: pass 2026-09-26
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 2026-09-27 with SymPy 1.14.0.