Derivative of \( \displaystyle \frac{\sqrt{\left(4 x + 1\right)^{2} + 1}}{4} \)
Problem 2.1257 · medium
Differentiate \( \displaystyle f(x) = \frac{\sqrt{\left(4 x + 1\right)^{2} + 1}}{4} \).
- \[ \frac{d}{d x} \frac{\sqrt{\left(4 x + 1\right)^{2} + 1}}{4} \]Start with the derivative of the function.✓ Proved
- \[ = \frac{\frac{d}{d x} \sqrt{\left(4 x + 1\right)^{2} + 1}}{4} \]constant-multiple rewritePull out the constant factor 1/4. Rewrite the square root as a power.✓ Proved
- \[ = \frac{\frac{d}{d x} \left(\left(4 x + 1\right)^{2} + 1\right)}{8 \sqrt{\left(4 x + 1\right)^{2} + 1}} \]chainApply the chain rule.✓ Proved
- \[ = \frac{\frac{d}{d x} \left(4 x + 1\right)^{2}}{8 \sqrt{\left(4 x + 1\right)^{2} + 1}} \]sumDifferentiate the sum inside the parenthesis.✓ Proved
- \[ = \frac{\left(8 x + 2\right) \frac{d}{d x} \left(4 x + 1\right)}{8 \sqrt{\left(4 x + 1\right)^{2} + 1}} \]powerApply the power rule and chain rule to the squared term.✓ Proved
- \[ = \frac{32 x + 8}{8 \sqrt{\left(4 x + 1\right)^{2} + 1}} \]derivative algebraDifferentiate the linear term. Multiply the constants.✓ Proved
- \[ = \frac{4 x + 1}{\sqrt{\left(4 x + 1\right)^{2} + 1}} \]simplifySimplify the final expression.✓ Proved
Answer \( \frac{4 x + 1}{\sqrt{\left(4 x + 1\right)^{2} + 1}} \)
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 (4*x + 1)**2 + 1 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (4*x + 1)**2 + 1 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (4*x + 1)**2 + 1 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (4*x + 1)**2 + 1 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (4*x + 1)**2 + 1 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (4*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 (4*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 5 drops the derivative of the constant term 1. The derivative of ((4*x+1)**2 + 1) should be 2*(4*x+1)*4 + 0, not just 2*(4*x+1)*4. This omission makes the result incorrect.qwen3.6:27b-mlx: fail (style) — Step 6 applies both the power rule and the chain rule simultaneously, violating the one-rule-per-step constraint. Step 5 is also imprecise, as it performs differentiation (derivative rule) rather than just applying the sum rule to split terms.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (style) 2026-09-29 — Step 6 applies both the power rule and the chain rule simultaneously, violating the one-rule-per-step constraint. Step 5 is also imprecise, as it performs differentiation (derivative rule) rather than just applying the sum rule to split terms.gpt-oss:20b: fail (error) 2026-09-29 — Step 5 drops the derivative of the constant term 1. The derivative of ((4*x+1)**2 + 1) should be 2*(4*x+1)*4 + 0, not just 2*(4*x+1)*4. This omission makes the result incorrect.qwen3.6:27b-mlx: fail (error) 2026-09-29 — Step 5 is labeled 'sum' but performs differentiation of the inner function, which is part of the chain rule application in Step 4; it does not apply the sum rule (differentiation of a sum) as a distinct operation separate from the chain rule context. Furthermore, Step 6 is labeled 'power' but applies both the power rule and the chain rule simultaneously to the term (4*x+1)^2, violating the one-rule-per-step constraint.gpt-oss:20b: pass 2026-09-29
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 2026-09-29 with SymPy 1.14.0.