Derivative of \( \displaystyle 5 \operatorname{atan}{\left(x + 2 \right)} \)
Problem 2.1113 · medium
Differentiate \( \displaystyle f(x) = 5 \operatorname{atan}{\left(x + 2 \right)} \).
- \[ \frac{d}{d x} 5 \operatorname{atan}{\left(x + 2 \right)} \]constant-multipleStart with the derivative of the function.✓ Proved
- \[ = 5 \frac{d}{d x} \operatorname{atan}{\left(x + 2 \right)} \]derivativeMove the constant outside the derivative.✓ Proved
- \[ = \frac{5 \frac{d}{d x} \left(x + 2\right)}{\left(x + 2\right)^{2} + 1} \]chainApply the chain rule to the arctangent function.✓ Proved
- \[ = \frac{5}{\left(x + 2\right)^{2} + 1} \]derivative algebraDifferentiate the inner function x + 2. Simplify the expression.✓ Proved
- \[ = \frac{5}{x^{2} + 4 x + 5} \]algebra simplifyExpand the denominator. Combine like terms.✓ Proved
Answer \( \frac{5}{\left(x + 2\right)^{2} + 1} \)
✓ 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.
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 (x + 2)**2 + 1 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (x + 2)**2 + 1 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (x + 2)**2 + 1 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (x + 2)**2 + 1 = 0 undefined where x**2 + 4*x + 5 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x**2 + 4*x + 5 = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where (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: fail (style) — Step 1 is labeled 'constant-multiple' but performs no operation; it merely restates the problem, violating the rule that each step must change one thing. Step 2 is labeled 'derivative' but performs the constant-multiple rule, mislabeling the operation.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (style) 2026-09-28 — Step 1 is labeled 'constant-multiple' but performs no operation; it merely restates the problem, violating the rule that each step must change one thing. Step 2 is labeled 'derivative' but performs the constant-multiple rule, mislabeling the operation.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: fail (style) 2026-09-28 — Step 1 is labeled 'constant-multiple' but performs no operation; it merely states the problem. Step 2 is labeled 'derivative' but performs the constant-multiple rule. The labels for steps 1 and 2 are swapped.gpt-oss:20b: fail (style) 2026-09-28 — Step 2 is labeled "derivative" but it only applies the constant‑multiple rule to pull the constant outside the derivative. The correct label should be "constant‑multiple". No mathematical errors are present, only a labeling defect.
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 2026-09-28 with SymPy 1.14.0.