Derivative of \( \displaystyle 5 \operatorname{atan}{\left(x - 3 \right)} \)
Problem 2.595 · medium
Differentiate \( \displaystyle f(x) = 5 \operatorname{atan}{\left(x - 3 \right)} \).
- \[ \frac{d}{d x} 5 \operatorname{atan}{\left(x - 3 \right)} \]constant-multipleApply the constant multiple rule.✓ Proved
- \[ = 5 \frac{d}{d x} \operatorname{atan}{\left(x - 3 \right)} \]chainApply the chain rule to the arctangent function.✓ Proved
- \[ = \frac{5 \frac{d}{d x} \left(x - 3\right)}{\left(x - 3\right)^{2} + 1} \]derivativeDifferentiate the inner function x - 3.✓ Proved
- \[ = \frac{5}{\left(x - 3\right)^{2} + 1} \]algebra algebraSimplify the expression. Simplify the fraction.✓ Proved
- \[ = \frac{5}{x^{2} - 6 x + 10} \]algebra simplifyExpand the squared term. Combine like terms.✓ Proved
Answer \( \frac{5}{\left(x - 3\right)^{2} + 1} \)
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. The reviewers disagree about how one step is explained; every verdict is in the receipt.
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 - 3)**2 + 1 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (x - 3)**2 + 1 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (x - 3)**2 + 1 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (x - 3)**2 + 1 = 0 undefined where x**2 - 6*x + 10 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x**2 - 6*x + 10 = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where (x - 3)**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 2 applies the chain rule but does not actually perform the differentiation of the arctangent; step 3 then applies two rules at once (derivative of atan and derivative of the inner function). Each step must change only one rule, so step 3 should be split into two separate steps.qwen3.6:27b-mlx: pass
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-21gpt-oss:20b: fail (error) 2026-09-21 — Step 2 applies the chain rule but does not actually perform the differentiation of the arctangent; step 3 then applies two rules at once (derivative of atan and derivative of the inner function). Each step must change only one rule, so step 3 should be split into two separate steps.qwen3.6:27b-mlx: pass 2026-09-21gpt-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.