Derivative of \( \displaystyle \sqrt{\left(x - 1\right)^{2} + 1} \)
Problem 2.1055 · medium
Differentiate \( \displaystyle f(x) = \sqrt{\left(x - 1\right)^{2} + 1} \).
- \[ \frac{d}{d x} \sqrt{\left(x - 1\right)^{2} + 1} \]derivative rewriteStart with the derivative of the function. Rewrite the square root as a fractional power.✓ Proved
- \[ = \frac{\frac{d}{d x} \left(\left(x - 1\right)^{2} + 1\right)}{2 \sqrt{\left(x - 1\right)^{2} + 1}} \]chainApply the chain rule.✓ Proved
- \[ = \frac{\frac{d}{d x} 1 + \frac{d}{d x} \left(x - 1\right)^{2}}{2 \sqrt{\left(x - 1\right)^{2} + 1}} \]sumDifferentiate the sum inside the parentheses.✓ Proved
- \[ = \frac{\left(2 x - 2\right) \frac{d}{d x} \left(x - 1\right)}{2 \sqrt{\left(x - 1\right)^{2} + 1}} \]powerApply the power rule to (x - 1)**2.✓ Proved
- \[ = \frac{2 x - 2}{2 \sqrt{\left(x - 1\right)^{2} + 1}} \]derivative simplifyDifferentiate the inner term (x - 1). Simplify the expression.✓ Proved
- \[ = \frac{x - 1}{\sqrt{\left(x - 1\right)^{2} + 1}} \]simplify simplifyCancel the 2 and 1/2. Rewrite with a positive exponent.✓ Proved
Answer \( \frac{x - 1}{\sqrt{\left(x - 1\right)^{2} + 1}} \)
✓ 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 (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 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (x - 1)**2 + 1 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (x - 1)**2 + 1 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (x - 1)**2 + 1 = 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: passqwen3.6:27b-mlx: pass — The solution correctly applies the chain rule, power rule, and sum rule in separate steps. All labels are valid and accurately describe the single operation performed in each step.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly applies the chain rule, power rule, and sum rule in separate steps. All labels are valid and accurately describe the single operation performed in each step.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly applies the chain rule, sum rule, and power rule in separate steps. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-09-28
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.