Derivative of \( \displaystyle \frac{5 \sqrt{\left(2 x + 1\right)^{2} + 1}}{2} \)
Problem 2.680 · medium
Differentiate \( \displaystyle f(x) = \frac{5 \sqrt{\left(2 x + 1\right)^{2} + 1}}{2} \).
- \[ \frac{d}{d x} \frac{5 \sqrt{\left(2 x + 1\right)^{2} + 1}}{2} \]Start with the derivative of the function.✓ Proved
- \[ = \frac{5 \frac{d}{d x} \sqrt{\left(2 x + 1\right)^{2} + 1}}{2} \]constant-multiple rewritePull out the constant factor. Rewrite the square root as a power.✓ Proved
- \[ = \frac{5 \frac{d}{d x} \left(\left(2 x + 1\right)^{2} + 1\right)}{4 \sqrt{\left(2 x + 1\right)^{2} + 1}} \]chainApply the chain rule.✓ Proved
- \[ = \frac{5 \left(\frac{d}{d x} 1 + \frac{d}{d x} \left(2 x + 1\right)^{2}\right)}{4 \sqrt{\left(2 x + 1\right)^{2} + 1}} \]algebraDistribute the power rule across the sum.✓ Proved
- \[ = \frac{5 \left(4 x + 2\right) \frac{d}{d x} \left(2 x + 1\right)}{4 \sqrt{\left(2 x + 1\right)^{2} + 1}} \]powerDifferentiate the squared term.✓ Proved
- \[ = \frac{5 \left(8 x + 4\right)}{4 \sqrt{\left(2 x + 1\right)^{2} + 1}} \]derivative algebraDifferentiate the inner linear term. Simplify the product of constants.✓ Proved
- \[ = \frac{10 x + 5}{\sqrt{\left(2 x + 1\right)^{2} + 1}} \]simplifySimplify the final expression.✓ Proved
Answer \( \frac{5 \left(2 x + 1\right)}{\sqrt{\left(2 x + 1\right)^{2} + 1}} \)
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. 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 (2*x + 1)**2 + 1 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (2*x + 1)**2 + 1 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (2*x + 1)**2 + 1 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (2*x + 1)**2 + 1 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (2*x + 1)**2 + 1 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (2*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 (2*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 (style) — Step 5 applies the sum rule to split the derivative of a sum, but labels it as "algebra". The correct label for that operation is "sum".qwen3.6:27b-mlx: fail (style) — Step 5 is labeled 'algebra' but performs differentiation (splitting the derivative of a sum), which should be labeled 'sum'. Step 6 is labeled 'power' but applies the chain rule to the squared term, which should be labeled 'chain'.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (style) 2026-09-21 — Step 5 is labeled 'algebra' but performs differentiation (splitting the derivative of a sum), which should be labeled 'sum'. Step 6 is labeled 'power' but applies the chain rule to the squared term, which should be labeled 'chain'.gpt-oss:20b: fail (style) 2026-09-21 — Step 5 applies the sum rule to split the derivative of a sum, but labels it as "algebra". The correct label for that operation is "sum".qwen3.6:27b-mlx: fail (error) 2026-09-21 — Step 5 is labeled 'algebra' but performs differentiation (splitting the derivative of a sum into a sum of derivatives), which should be labeled 'sum'. Additionally, Step 5 applies the linearity of the derivative operator, but the label 'algebra' is incorrect for a differentiation step; however, the primary defect is that Step 5 combines the distribution of the outer derivative (from step 4) with the splitting of the inner derivative, effectively applying the 'sum' rule inside the derivative context without a proper label, or rather, the label 'algebra' is used for a step that is fundamentally about the linearity of differentiation (sum rule). Wait, looking closer: Step 4 applies chain rule to get Derivative(inner). Step 5 expands Derivative(inner) into Derivative(term1) + Derivative(term2). This is the 'sum' rule. Labeling it 'algebra' is a defect because 'algebra' is for non-differentiation manipulations. The note says 'Distribute the power rule across the sum', which is mathematically false (the power rule was already applied in step 4 to the outer function; step 5 is just splitting the derivative of the inner function). This is misleading/wrong.gpt-oss:20b: fail (error) 2026-09-21 — Step 5 applies two rules at once—distributing the derivative over the sum requires the sum rule, but the step is labeled only as algebra. This violates the one‑rule‑per‑step rule and mislabels the operation.
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.