Derivative of \( \displaystyle - \frac{\sqrt{\left(5 x - 1\right)^{2} + 1}}{5} \)
Problem 2.999 · medium
Differentiate \( \displaystyle f(x) = - \frac{\sqrt{\left(5 x - 1\right)^{2} + 1}}{5} \).
- \[ \frac{d}{d x} \left(- \frac{\sqrt{\left(5 x - 1\right)^{2} + 1}}{5}\right) \]Start with the derivative of the function.✓ Proved
- \[ = - \frac{\frac{d}{d x} \sqrt{\left(5 x - 1\right)^{2} + 1}}{5} \]constant-multiple rewritePull out the constant factor. Rewrite the square root as a fractional exponent.✓ Proved
- \[ = - \frac{\frac{d}{d x} \left(\left(5 x - 1\right)^{2} + 1\right)}{10 \sqrt{\left(5 x - 1\right)^{2} + 1}} \]chainApply the chain rule.✓ Proved
- \[ = - \frac{\frac{d}{d x} \left(5 x - 1\right)^{2}}{10 \sqrt{\left(5 x - 1\right)^{2} + 1}} \]sumDifferentiate the sum inside the parenthesis.✓ Proved
- \[ = - \frac{\left(10 x - 2\right) \frac{d}{d x} \left(5 x - 1\right)}{10 \sqrt{\left(5 x - 1\right)^{2} + 1}} \]powerApply the power rule to the squared term.✓ Proved
- \[ = - \frac{50 x - 10}{10 \sqrt{\left(5 x - 1\right)^{2} + 1}} \]derivative constant-multipleDifferentiate the linear term. Multiply the constants.✓ Proved
- \[ = \frac{1 - 5 x}{\sqrt{\left(5 x - 1\right)^{2} + 1}} \]simplifySimplify the final expression.✓ Proved
Answer \( \frac{1 - 5 x}{\sqrt{\left(5 x - 1\right)^{2} + 1}} \)
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 (5*x - 1)**2 + 1 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (5*x - 1)**2 + 1 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (5*x - 1)**2 + 1 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (5*x - 1)**2 + 1 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (5*x - 1)**2 + 1 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (5*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 (5*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 applies the sum rule but omits the derivative of the constant term 1, effectively differentiating only (5*x-1)**2. This is a missing application of the derivative rule and makes the step incorrect.qwen3.6:27b-mlx: pass
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-27gpt-oss:20b: fail (error) 2026-09-27 — Step 5 applies the sum rule but omits the derivative of the constant term 1, effectively differentiating only (5*x-1)**2. This is a missing application of the derivative rule and makes the step incorrect.qwen3.6:27b-mlx: pass 2026-09-27gpt-oss:20b: fail (error) 2026-09-27 — Step 5 applies the sum rule but omits the derivative of the constant term 1, effectively applying two rules at once (differentiating the sum and ignoring the constant).
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 2026-09-27 with SymPy 1.14.0.