Derivative of \( \displaystyle \frac{\left(5 x + 1\right)^{2}}{5 x + 2} \)
Problem 2.320 · hard
Differentiate \( \displaystyle f(x) = \frac{\left(5 x + 1\right)^{2}}{5 x + 2} \).
- \[ \frac{d}{d x} \frac{\left(5 x + 1\right)^{2}}{5 x + 2} \]derivative rewriteStart with the derivative of the function. Rewrite the quotient using a negative exponent.✓ Proved
- \[ = \left(5 x + 1\right)^{2} \frac{d}{d x} \frac{1}{5 x + 2} + \frac{\frac{d}{d x} \left(5 x + 1\right)^{2}}{5 x + 2} \]productApply the product rule.✓ Proved
- \[ = \left(5 x + 1\right)^{2} \frac{d}{d x} \frac{1}{5 x + 2} + \frac{\left(10 x + 2\right) \frac{d}{d x} \left(5 x + 1\right)}{5 x + 2} \]powerApply the chain rule to the first term.✓ Proved
- \[ = \left(5 x + 1\right)^{2} \frac{d}{d x} \frac{1}{5 x + 2} + \frac{50 x + 10}{5 x + 2} \]derivative constant-multipleDifferentiate the inner function 5*x + 1. Multiply the constants.✓ Proved
- \[ = - \frac{\left(5 x + 1\right)^{2} \frac{d}{d x} \left(5 x + 2\right)}{\left(5 x + 2\right)^{2}} + \frac{50 x + 10}{5 x + 2} \]powerApply the chain rule to the second term.✓ Proved
- \[ = - \frac{5 \left(5 x + 1\right)^{2}}{\left(5 x + 2\right)^{2}} + \frac{50 x + 10}{5 x + 2} \]derivative constant-multipleDifferentiate the inner function 5*x + 2. Simplify the signs and constants.✓ Proved
- \[ = \frac{- 5 \left(5 x + 1\right)^{2} + \left(5 x + 2\right) \left(50 x + 10\right)}{\left(5 x + 2\right)^{2}} \]algebraFind a common denominator.✓ Proved
- \[ = \frac{125 x^{2} + 100 x + 15}{\left(5 x + 2\right)^{2}} \]algebra algebra simplifyExpand the terms in the numerator. Distribute the constants. Combine like terms to get the final answer.✓ Proved
Answer \( \frac{5 \left(5 x + 1\right) \left(5 x + 3\right)}{\left(5 x + 2\right)^{2}} \)
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 undefined where 5*x + 2 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 5*x + 2 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 5*x + 2 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 5*x + 2 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 5*x + 2 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 5*x + 2 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 5*x + 2 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 5*x + 2 = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 5*x + 2 = 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 5*x + 2 = 0 |
| 12 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 5*x + 2 = 0 |
| 13 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 5*x + 2 = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where 5*x + 2 = 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) — Steps 4 and 7 are labeled 'power' but the notes explicitly state 'Apply the chain rule'. The label should be 'chain' to match the operation described and performed (differentiating a composite function). Additionally, Step 9 combines sign simplification and constant multiplication, which is slightly ambiguous but acceptable under 'constant-multiple'; however, the labeling inconsistency in steps 4 and 7 is a clear defect.
Every verdict on record (12)
qwen3.6:27b-mlx: fail (style) 2026-09-20 — Steps 4 and 7 are labeled 'power' but the notes explicitly state 'Apply the chain rule'. The label should be 'chain' to match the operation described and performed (differentiating a composite function). Additionally, Step 9 combines sign simplification and constant multiplication, which is slightly ambiguous but acceptable under 'constant-multiple'; however, the labeling inconsistency in steps 4 and 7 is a clear defect.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: inconclusive 2026-09-20 — reviewer response could not be parsed: { "verdict": "fail", "severity": "style", "notes": "Steps 4 and 7 are labeled 'power' but the notes explicitly state 'Apply the chain rule'. While the power rule is applied, the step involves digpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: fail (style) 2026-09-19 — Steps 4 and 7 are labeled 'power' but the notes explicitly state 'Apply the chain rule'. The label 'power' is insufficient because the step applies the chain rule to the inner function, which is a distinct rule from the power rule. Additionally, Step 9 combines sign simplification and constant multiplication, which is acceptable under 'constant-multiple', but the primary issue is the mislabeling of the chain rule steps.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: fail (style) 2026-09-19 — Steps 4 and 7 are labeled 'power' but the notes explicitly state 'Apply the chain rule'. The label 'chain' exists in the vocabulary and is the correct rule for differentiating a composite function like (5*x+1)^2 or (5*x+2)^(-1). Using 'power' is misleading because the power rule alone does not account for the inner derivative, which is the primary operation being performed in these steps.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19gpt-oss:20b: fail 2026-09-17 — Step 4’s note claims the chain rule is applied, but at that point only the power rule has been used; the chain rule is applied in the next step. This misleading note could confuse a student.deepseek-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.