Derivative of \( \displaystyle \frac{\sqrt{\left(2 x - 3\right)^{2} + 1}}{2} \)
Problem 2.889 · medium
Differentiate \( \displaystyle f(x) = \frac{\sqrt{\left(2 x - 3\right)^{2} + 1}}{2} \).
- \[ \frac{d}{d x} \frac{\sqrt{\left(2 x - 3\right)^{2} + 1}}{2} \]Start with the derivative of the function.✓ Proved
- \[ = \frac{\frac{d}{d x} \sqrt{\left(2 x - 3\right)^{2} + 1}}{2} \]constant-multiple rewritePull out the constant factor 1/2. Rewrite the square root as a fractional power.✓ Proved
- \[ = \frac{\frac{d}{d x} \left(\left(2 x - 3\right)^{2} + 1\right)}{4 \sqrt{\left(2 x - 3\right)^{2} + 1}} \]chainApply the chain rule.✓ Proved
- \[ = \frac{\frac{d}{d x} \left(2 x - 3\right)^{2}}{4 \sqrt{\left(2 x - 3\right)^{2} + 1}} \]sumDifferentiate the sum inside the parentheses.✓ Proved
- \[ = \frac{\left(4 x - 6\right) \frac{d}{d x} \left(2 x - 3\right)}{4 \sqrt{\left(2 x - 3\right)^{2} + 1}} \]powerApply the power rule to the squared term.✓ Proved
- \[ = \frac{8 x - 12}{4 \sqrt{\left(2 x - 3\right)^{2} + 1}} \]derivative algebraDifferentiate the linear term. Multiply the constants.✓ Proved
- \[ = \frac{2 x - 3}{\sqrt{\left(2 x - 3\right)^{2} + 1}} \]simplifySimplify the expression to its final form.✓ Proved
Answer \( \frac{2 x - 3}{\sqrt{\left(2 x - 3\right)^{2} + 1}} \)
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 (2*x - 3)**2 + 1 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (2*x - 3)**2 + 1 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (2*x - 3)**2 + 1 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (2*x - 3)**2 + 1 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (2*x - 3)**2 + 1 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (2*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 (2*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 5 applies the sum rule but omits the derivative of the constant term 1. The derivative of 1 is 0, so the step should either include it explicitly or note that it vanishes. This omission makes the step mathematically incorrect.qwen3.6:27b-mlx: pass
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-26gpt-oss:20b: fail (error) 2026-09-26 — Step 5 applies the sum rule but omits the derivative of the constant term 1. The derivative of 1 is 0, so the step should either include it explicitly or note that it vanishes. This omission makes the step mathematically incorrect.qwen3.6:27b-mlx: pass 2026-09-26gpt-oss:20b: fail (error) 2026-09-26 — Step 5 drops the derivative of the constant term +1. The chain rule produced a factor of ((2*x-3)**2+1)**(-1/2) times Derivative((2*x-3)**2+1,x), but the solution replaces it with Derivative((2*x-3)**2,x), effectively applying the sum rule incorrectly by omitting the derivative of the constant. This is a mathematical error.
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.