Derivative of \( \displaystyle 5 \operatorname{atan}{\left(x - 1 \right)} \)
Problem 2.1139 · medium Video
Differentiate \( \displaystyle f(x) = 5 \operatorname{atan}{\left(x - 1 \right)} \).
- \[ \frac{d}{d x} 5 \operatorname{atan}{\left(x - 1 \right)} \]constant-multipleApply the constant multiple rule.✓ Proved
- \[ = 5 \frac{d}{d x} \operatorname{atan}{\left(x - 1 \right)} \]chainApply the chain rule to the inner function.✓ Proved
- \[ = \frac{5 \frac{d}{d x} \left(x - 1\right)}{\left(x - 1\right)^{2} + 1} \]derivativeDifferentiate the inner function x - 1.✓ Proved
- \[ = \frac{5}{\left(x - 1\right)^{2} + 1} \]algebra algebraSimplify the expression. Expand the denominator.✓ Proved
- \[ = \frac{5}{x^{2} - 2 x + 2} \]simplify simplifyCombine like terms. Final simplified form.✓ Proved
Answer \( \frac{5}{\left(x - 1\right)^{2} + 1} \)
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. A reviewer's objection to the explanation stands; see the receipt. 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 (x - 1)**2 + 1 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (x - 1)**2 + 1 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (x - 1)**2 + 1 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (x - 1)**2 + 1 = 0 undefined where x**2 - 2*x + 2 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x**2 - 2*x + 2 = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where (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: fail (error) — Step 3 applies two rules at once: it expands the derivative of atan(x‑1) into 1/(1+(x‑1)²)·Derivative(x‑1,x) and simultaneously differentiates the inner function x‑1. Each rule must be applied in a separate step, and the label should reflect the correct rule (e.g., chain or derivative).qwen3.6:27b-mlx: fail (error) — Step 2 is labeled 'chain' but performs no differentiation; it merely restates the expression from Step 1. The chain rule is actually applied in Step 3, which is incorrectly labeled 'derivative' (it should be 'chain' or 'derivative' depending on if it's the outer or inner, but Step 2 is definitely not a valid application of the chain rule as a transformation step).
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-03 — Step 2 is labeled 'chain' but performs no differentiation; it merely restates the expression from Step 1. The chain rule is actually applied in Step 3, which is incorrectly labeled 'derivative' (it should be 'chain' or 'derivative' depending on if it's the outer or inner, but Step 2 is definitely not a valid application of the chain rule as a transformation step).gpt-oss:20b: fail (error) 2026-10-03 — Step 3 applies two rules at once: it expands the derivative of atan(x‑1) into 1/(1+(x‑1)²)·Derivative(x‑1,x) and simultaneously differentiates the inner function x‑1. Each rule must be applied in a separate step, and the label should reflect the correct rule (e.g., chain or derivative).qwen3.6:27b-mlx: pass 2026-09-28gpt-oss:20b: fail (error) 2026-09-28 — Step 2 incorrectly labels a constant‑multiple step as "chain". Step 3 applies two rules at once (the derivative of atan and the derivative of the inner function), violating the one‑rule‑per‑step rule.qwen3.6:27b-mlx: fail (style) 2026-09-28 — Step 2 is labeled 'chain' but only applies the constant multiple rule to pull out the 5; the chain rule is not applied until Step 3. Step 2 should be labeled 'constant-multiple'.gpt-oss:20b: fail (error) 2026-09-28 — Step 3 applies two rules at once: it uses the derivative of atan (which already incorporates the chain rule) and then separately differentiates the inner function x‑1. This violates the one‑rule‑per‑step rule and the label "derivative" is insufficient to describe the combined operations.
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 2026-10-03 with SymPy 1.14.0.