Derivative of \( \displaystyle 2 \sqrt{\left(x - 3\right)^{2} + 1} \)
Problem 2.900 · medium
Differentiate \( \displaystyle f(x) = 2 \sqrt{\left(x - 3\right)^{2} + 1} \).
- \[ \frac{d}{d x} 2 \sqrt{\left(x - 3\right)^{2} + 1} \]Start with the derivative of the function.✓ Proved
- \[ = 2 \frac{d}{d x} \sqrt{\left(x - 3\right)^{2} + 1} \]constant-multiple rewritePull the constant out of the derivative. Rewrite the square root as a fractional exponent.✓ Proved
- \[ = \frac{\frac{d}{d x} \left(\left(x - 3\right)^{2} + 1\right)}{\sqrt{\left(x - 3\right)^{2} + 1}} \]chain algebraApply the chain rule. Simplify the constant multiplication.✓ Proved
- \[ = \frac{\frac{d}{d x} \left(x - 3\right)^{2}}{\sqrt{\left(x - 3\right)^{2} + 1}} \]sumDifferentiate the sum inside the parentheses.✓ Proved
- \[ = \frac{\left(2 x - 6\right) \frac{d}{d x} \left(x - 3\right)}{\sqrt{\left(x - 3\right)^{2} + 1}} \]powerApply the power rule to (x - 3)**2.✓ Proved
- \[ = \frac{2 x - 6}{\sqrt{\left(x - 3\right)^{2} + 1}} \]derivative algebra simplifyDifferentiate the inner term (x - 3). Simplify the expression. Rewrite with a positive exponent in the denominator.✓ Proved
Answer \( \frac{2 \left(x - 3\right)}{\sqrt{\left(x - 3\right)^{2} + 1}} \)
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 |
| 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 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (x - 3)**2 + 1 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (x - 3)**2 + 1 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (x - 3)**2 + 1 = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (x - 3)**2 + 1 = 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: passqwen3.6:27b-mlx: fail (error) — Step 6 is labeled 'sum' but applies the derivative operator to the term (x-3)^2 while dropping the derivative of the constant 1. The label 'sum' implies applying the sum rule to split the derivative, but the step performs differentiation of a specific term, which should be labeled 'derivative' or handled via 'constant' for the 1. More critically, the step effectively differentiates (x-3)^2 + 1 into Derivative((x-3)^2, x) + 0, but the label 'sum' is not the standard label for differentiating a sum term-by-term in this context (usually 'derivative' covers the operation on the whole expression or 'sum' splits it into two derivatives). However, the primary defect is that Step 6 claims to apply the 'sum' rule but only shows the derivative of the first term, implicitly handling the constant derivative as 0 without showing the split or labeling it 'derivative'. A step labeled 'sum' should typically show d(u+v) = du + dv. Here it jumps to just du. This is a granularity/labeling error where the rule applied (differentiation of a sum) is not correctly represented by the label 'sum' which usually denotes the linearity property d(u+v)=du+dv, whereas the step performed the differentiation of the variable part. Actually, looking closer, Step 6 goes from `... * Derivative((x - 3)**2 + 1, x)` to `... * Derivative((x - 3)**2, x)`. This step removes the `+ 1` inside the derivative. This is an application of the derivative of a constant being zero, combined with the sum rule. It applies two concepts (sum rule and constant derivative) or just the derivative of a sum. If it's just 'sum', it should be `Derivative((x-3)^2, x) + Derivative(1, x)`. It skips to just the first term. This is a multi-rule step or an incomplete application. Let's look at Step 7. Step 7 applies power rule to `(x-3)^2`. This is fine. Step 6 is the issue. It changes `Derivative((x - 3)**2 + 1, x)` to `Derivative((x - 3)**2, x)`. This is not a standard single rule from the list. It's `sum` + `constant`. Or it's `derivative` if we consider the whole thing. But `derivative` usually unfolds known forms. `sum` splits. This step does neither cleanly. It effectively says `d(u+c) = du`. This requires `sum` and `constant`. Two rules.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-26 — Step 6 is labeled 'sum' but applies the derivative operator to the term (x-3)^2 while dropping the derivative of the constant 1. The label 'sum' implies applying the sum rule to split the derivative, but the step performs differentiation of a specific term, which should be labeled 'derivative' or handled via 'constant' for the 1. More critically, the step effectively differentiates (x-3)^2 + 1 into Derivative((x-3)^2, x) + 0, but the label 'sum' is not the standard label for differentiating a sum term-by-term in this context (usually 'derivative' covers the operation on the whole expression or 'sum' splits it into two derivatives). However, the primary defect is that Step 6 claims to apply the 'sum' rule but only shows the derivative of the first term, implicitly handling the constant derivative as 0 without showing the split or labeling it 'derivative'. A step labeled 'sum' should typically show d(u+v) = du + dv. Here it jumps to just du. This is a granularity/labeling error where the rule applied (differentiation of a sum) is not correctly represented by the label 'sum' which usually denotes the linearity property d(u+v)=du+dv, whereas the step performed the differentiation of the variable part. Actually, looking closer, Step 6 goes from `... * Derivative((x - 3)**2 + 1, x)` to `... * Derivative((x - 3)**2, x)`. This step removes the `+ 1` inside the derivative. This is an application of the derivative of a constant being zero, combined with the sum rule. It applies two concepts (sum rule and constant derivative) or just the derivative of a sum. If it's just 'sum', it should be `Derivative((x-3)^2, x) + Derivative(1, x)`. It skips to just the first term. This is a multi-rule step or an incomplete application. Let's look at Step 7. Step 7 applies power rule to `(x-3)^2`. This is fine. Step 6 is the issue. It changes `Derivative((x - 3)**2 + 1, x)` to `Derivative((x - 3)**2, x)`. This is not a standard single rule from the list. It's `sum` + `constant`. Or it's `derivative` if we consider the whole thing. But `derivative` usually unfolds known forms. `sum` splits. This step does neither cleanly. It effectively says `d(u+c) = du`. This requires `sum` and `constant`. Two rules.gpt-oss:20b: pass 2026-09-26qwen3.6:27b-mlx: pass 2026-09-26gpt-oss:20b: fail (error) 2026-09-26 — Step 6 applies the sum rule but omits the derivative of the constant term +1. The correct derivative of (x-3)**2 + 1 is 2*(x-3) + 0, not just 2*(x-3).
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.