Derivative of \( \displaystyle \frac{5 \sqrt{\left(4 x - 1\right)^{2} + 1}}{4} \)
Problem 2.180 · medium
Differentiate \( \displaystyle f(x) = \frac{5 \sqrt{\left(4 x - 1\right)^{2} + 1}}{4} \).
- \[ \frac{d}{d x} \frac{5 \sqrt{\left(4 x - 1\right)^{2} + 1}}{4} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{5 \frac{d}{d x} \sqrt{\left(4 x - 1\right)^{2} + 1}}{4} \]constant-multiple rewritePull out the constant factor. Rewrite the square root as a power.✓ Proved
- \[ = \frac{5 \frac{d}{d x} \left(\left(4 x - 1\right)^{2} + 1\right)}{8 \sqrt{\left(4 x - 1\right)^{2} + 1}} \]chain algebraApply the chain rule. Simplify the constant multiplication.✓ Proved
- \[ = \frac{5 \frac{d}{d x} \left(4 x - 1\right)^{2}}{8 \sqrt{\left(4 x - 1\right)^{2} + 1}} \]sumApply the sum rule to the derivative.✓ Proved
- \[ = \frac{5 \left(8 x - 2\right) \frac{d}{d x} \left(4 x - 1\right)}{8 \sqrt{\left(4 x - 1\right)^{2} + 1}} \]chainApply the chain rule to the squared term.✓ Proved
- \[ = \frac{5 \left(32 x - 8\right)}{8 \sqrt{\left(4 x - 1\right)^{2} + 1}} \]derivative algebraDifferentiate the inner linear function. Multiply the constants in the derivative term.✓ Proved
- \[ = \frac{20 x - 5}{\sqrt{\left(4 x - 1\right)^{2} + 1}} \]algebra simplifySimplify the final expression by canceling 8/8. Rewrite with a positive exponent in the denominator.✓ Proved
Answer \( \frac{5 \left(4 x - 1\right)}{\sqrt{\left(4 x - 1\right)^{2} + 1}} \)
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 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where (4*x - 1)**2 + 1 = 0 |
| 11 | ✓ 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: passdeepseek-r1:70b: passqwen3.6:27b-mlx: fail (style) — Step 6 is labeled 'sum' but effectively applies the rule that the derivative of a constant is zero (a 'derivative' or 'constant' step), not the sum rule for splitting derivatives. The sum rule would split d/dx(u + v) into d/dx(u) + d/dx(v); this step just drops the constant term.
Every verdict on record (15)
qwen3.6:27b-mlx: fail (style) 2026-09-20 — Step 6 is labeled 'sum' but effectively applies the rule that the derivative of a constant is zero (a 'derivative' or 'constant' step), not the sum rule for splitting derivatives. The sum rule would split d/dx(u + v) into d/dx(u) + d/dx(v); this step just drops the constant term.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-20gpt-oss:20b: fail (error) 2026-09-20 — Step 6 applies the sum rule but omits the derivative of the constant term 1, resulting in an incomplete derivative. The correct derivative of (4*x-1)**2 + 1 should include 0 for the constant, but it is missing, making the step mathematically incorrect.qwen3.6:27b-mlx: pass 2026-09-19deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: pass 2026-09-19 — The solution correctly applies the rules in the specified vocabulary. Each step changes only one aspect of the expression, and the labels accurately reflect the operations performed.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: fail (error) 2026-09-19 — Step 6 applies the sum rule but then ignores the derivative of the constant term +1, effectively dropping a zero. The derivative of (4*x-1)**2 + 1 should include 0, but the subsequent steps omit it, leading to a correct result by accident but a logically incorrect step.qwen3.6:27b-mlx: fail (style) 2026-09-18 — Step 6 is labeled 'sum' but effectively applies the constant rule (derivative of 1 is 0) to remove a term; 'sum' is not the primary rule applied here, and the step skips the explicit differentiation of the constant term, making the label inaccurate for the transformation shown.deepseek-r1:70b: pass 2026-09-18gpt-oss:20b: pass 2026-09-18gpt-oss:20b: pass 2026-09-17deepseek-r1:70b: pass 2026-09-17
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.