Derivative of \( \displaystyle \frac{5 x^{2}}{4} + \frac{x}{2} - \left(\frac{5 x^{2}}{2} + x + \frac{1}{10}\right) \ln{\left(5 x + 1 \right)} \)
Problem 2.2382 · hard Video
Differentiate \( \displaystyle f(x) = \frac{5 x^{2}}{4} + \frac{x}{2} - \left(\frac{5 x^{2}}{2} + x + \frac{1}{10}\right) \ln{\left(5 x + 1 \right)} \).
- \[ \frac{d}{d x} \left(\frac{5 x^{2}}{4} + \frac{x}{2} - \left(\frac{5 x^{2}}{2} + x + \frac{1}{10}\right) \ln{\left(5 x + 1 \right)}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{5 x^{2}}{4} - \frac{d}{d x} \left(\frac{5 x^{2}}{2} + x + \frac{1}{10}\right) \ln{\left(5 x + 1 \right)} \]sumApply the sum rule.✓ Proved
- \[ = - \left(\frac{5 x^{2}}{2} + x + \frac{1}{10}\right) \frac{d}{d x} \ln{\left(5 x + 1 \right)} - \ln{\left(5 x + 1 \right)} \frac{d}{d x} \left(\frac{5 x^{2}}{2} + x + \frac{1}{10}\right) + \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{5 x^{2}}{4} \]productApply the product rule to the second term.✓ Proved
- \[ = - \left(\frac{5 x^{2}}{2} + x + \frac{1}{10}\right) \frac{d}{d x} \ln{\left(5 x + 1 \right)} - \left(\frac{d}{d x} \frac{1}{10} + \frac{d}{d x} x + \frac{d}{d x} \frac{5 x^{2}}{2}\right) \ln{\left(5 x + 1 \right)} + \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{5 x^{2}}{4} \]sumDifferentiate the sum inside the parenthesis.✓ Proved
- \[ = - \left(\frac{d}{d x} x + \frac{d}{d x} \frac{5 x^{2}}{2}\right) \ln{\left(5 x + 1 \right)} - \left(\frac{5 x^{2}}{2} + x + \frac{1}{10}\right) \frac{d}{d x} \ln{\left(5 x + 1 \right)} + \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{5 x^{2}}{4} \]constantThe derivative of the constant 1/10 is 0.✓ Proved
- \[ = - \left(5 x + 1\right) \ln{\left(5 x + 1 \right)} - \left(\frac{5 x^{2}}{2} + x + \frac{1}{10}\right) \frac{d}{d x} \ln{\left(5 x + 1 \right)} + \frac{d}{d x} \frac{x}{2} + \frac{d}{d x} \frac{5 x^{2}}{4} \]derivativeCompute the derivatives of the polynomial and identity terms.✓ Proved
- \[ = \frac{5 x}{2} - \left(5 x + 1\right) \ln{\left(5 x + 1 \right)} - \left(\frac{5 x^{2}}{2} + x + \frac{1}{10}\right) \frac{d}{d x} \ln{\left(5 x + 1 \right)} + \frac{1}{2} \]derivative algebraDifferentiate the remaining terms. Simplify the first term.✓ Proved
- \[ = \frac{5 x}{2} - \left(5 x + 1\right) \ln{\left(5 x + 1 \right)} + \frac{1}{2} - \frac{5 \left(\frac{5 x^{2}}{2} + x + \frac{1}{10}\right)}{5 x + 1} \]chain algebraApply the chain rule to the logarithm. Distribute the 5 in the last term's derivative.✓ Proved
- \[ = \frac{5 x}{2} - \left(5 x + 1\right) \ln{\left(5 x + 1 \right)} + \frac{1}{2} - \frac{\frac{25 x^{2}}{2} + 5 x + \frac{1}{2}}{5 x + 1} \]algebraMultiply the numerator by 5.✓ Proved
- \[ = \frac{5 x}{2} - \left(5 x + 1\right) \ln{\left(5 x + 1 \right)} + \frac{1}{2} - \frac{25 x^{2} + 10 x + 1}{10 x + 2} \]algebraFactor out 1/2 from the numerator.✓ Proved
- \[ = \frac{5 x}{2} - \frac{\left(5 x + 1\right)^{2}}{10 x + 2} - \left(5 x + 1\right) \ln{\left(5 x + 1 \right)} + \frac{1}{2} \]algebra algebraRecognize the perfect square trinomial. Simplify the fraction by canceling (5*x + 1).✓ Proved
- \[ = - \left(5 x + 1\right) \ln{\left(5 x + 1 \right)} \]simplify algebraSimplify the remaining fraction. Distribute the division by 2.✓ Proved
- \[ = \left(- 5 x - 1\right) \ln{\left(5 x + 1 \right)} \]simplifyCombine like terms to reach the final answer.✓ Proved
Answer \( \left(- 5 x - 1\right) \ln{\left(5 x + 1 \right)} \)
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. How we verify
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 log is undefined for non-positive arguments |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x + 1 = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x + 1 = 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x + 1 = 0 |
| 12 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 5*x + 1 = 0 undefined where 10*x + 2 = 0 |
| 13 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 10*x + 2 = 0 |
| 14 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 10*x + 2 = 0 |
| 15 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 10*x + 2 = 0 |
| 16 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 17 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 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 14 incorrectly rewrites (5*x+1)**2/(2*(5*x+1)) as (5*x+1)/2*(5*x+1). The correct simplification is (5*x+1)/2. This single step applies two algebraic manipulations at once and misstates the result, leading to an incorrect final expression.qwen3.6:27b-mlx: pass — The solution correctly applies differentiation rules and algebraic simplifications. Each step adheres to the single-change constraint and uses valid labels from the fixed vocabulary.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly applies differentiation rules and algebraic simplifications. Each step adheres to the single-change constraint and uses valid labels from the fixed vocabulary.gpt-oss:20b: fail (error) 2026-10-18 — Step 14 incorrectly rewrites (5*x+1)**2/(2*(5*x+1)) as (5*x+1)/2*(5*x+1). The correct simplification is (5*x+1)/2. This single step applies two algebraic manipulations at once and misstates the result, leading to an incorrect final expression.qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly applies differentiation rules and algebraic simplifications. Each step isolates a single operation, and the labels accurately reflect the rules applied.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: fail (style) 2026-10-15 — Step 6 applies the derivative rule to multiple terms (5*x**2/2 and x) simultaneously, violating the one-change-per-step constraint. Step 14 is algebraically incorrect as written (it multiplies by (5*x+1) instead of just canceling), although the transition to Step 15 is correct.gpt-oss:20b: pass 2026-10-15
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 2026-10-18 with SymPy 1.14.0.