Derivative of \( \displaystyle \frac{x^{2}}{4} + x - \left(\frac{x^{2}}{2} + 2 x + 2\right) \ln{\left(x + 2 \right)} \)
Problem 2.2378 · hard Video
Differentiate \( \displaystyle f(x) = \frac{x^{2}}{4} + x - \left(\frac{x^{2}}{2} + 2 x + 2\right) \ln{\left(x + 2 \right)} \).
- \[ \frac{d}{d x} \left(\frac{x^{2}}{4} + x - \left(\frac{x^{2}}{2} + 2 x + 2\right) \ln{\left(x + 2 \right)}\right) \]sumStart with the derivative of the entire function.✓ Proved
- \[ = \frac{d}{d x} x + \frac{d}{d x} \frac{x^{2}}{4} - \frac{d}{d x} \left(\frac{x^{2}}{2} + 2 x + 2\right) \ln{\left(x + 2 \right)} \]sumApply the sum rule to separate the terms.✓ Proved
- \[ = - \left(\frac{x^{2}}{2} + 2 x + 2\right) \frac{d}{d x} \ln{\left(x + 2 \right)} - \ln{\left(x + 2 \right)} \frac{d}{d x} \left(\frac{x^{2}}{2} + 2 x + 2\right) + \frac{d}{d x} x + \frac{d}{d x} \frac{x^{2}}{4} \]productApply the product rule to the third term.✓ Proved
- \[ = - \left(\frac{x^{2}}{2} + 2 x + 2\right) \frac{d}{d x} \ln{\left(x + 2 \right)} - \left(\frac{d}{d x} 2 + \frac{d}{d x} 2 x + \frac{d}{d x} \frac{x^{2}}{2}\right) \ln{\left(x + 2 \right)} + \frac{d}{d x} x + \frac{d}{d x} \frac{x^{2}}{4} \]sumApply the sum rule to the first part of the product rule.✓ Proved
- \[ = - \left(\frac{d}{d x} 2 x + \frac{d}{d x} \frac{x^{2}}{2}\right) \ln{\left(x + 2 \right)} - \left(\frac{x^{2}}{2} + 2 x + 2\right) \frac{d}{d x} \ln{\left(x + 2 \right)} + \frac{d}{d x} x + \frac{d}{d x} \frac{x^{2}}{4} \]constant algebraThe derivative of the constant 2 is 0. Simplify the expression inside the parentheses.✓ Proved
- \[ = - \left(2 \frac{d}{d x} x + \frac{d}{d x} \frac{x^{2}}{2}\right) \ln{\left(x + 2 \right)} - \left(\frac{x^{2}}{2} + 2 x + 2\right) \frac{d}{d x} \ln{\left(x + 2 \right)} + \frac{d}{d x} x + \frac{d}{d x} \frac{x^{2}}{4} \]constant-multipleFactor out the constant 2 from the derivative term.✓ Proved
- \[ = - \left(2 \frac{d}{d x} x + \frac{\frac{d}{d x} x^{2}}{2}\right) \ln{\left(x + 2 \right)} - \left(\frac{x^{2}}{2} + 2 x + 2\right) \frac{d}{d x} \ln{\left(x + 2 \right)} + \frac{d}{d x} x + \frac{d}{d x} \frac{x^{2}}{4} \]constant-multipleFactor out 1/2 from the derivative term.✓ Proved
- \[ = - \left(x \frac{d}{d x} x + 2 \frac{d}{d x} x\right) \ln{\left(x + 2 \right)} - \left(\frac{x^{2}}{2} + 2 x + 2\right) \frac{d}{d x} \ln{\left(x + 2 \right)} + \frac{d}{d x} x + \frac{d}{d x} \frac{x^{2}}{4} \]power algebraApply the power rule to x**2. Simplify the product 1/2 * 2 * x.✓ Proved
- \[ = - \left(x + 2\right) \ln{\left(x + 2 \right)} \frac{d}{d x} x - \left(\frac{x^{2}}{2} + 2 x + 2\right) \frac{d}{d x} \ln{\left(x + 2 \right)} + \frac{d}{d x} x + \frac{d}{d x} \frac{x^{2}}{4} \]algebraFactor out the common term Derivative(x, x).✓ Proved
- \[ = - \left(x + 2\right) \ln{\left(x + 2 \right)} + \frac{d}{d x} x + \frac{d}{d x} \frac{x^{2}}{4} - \frac{\frac{x^{2}}{2} + 2 x + 2}{x + 2} \]logarithmic algebraApply the derivative rule for log(x + 2). Simplify the expression by multiplying by 1.✓ Proved
- \[ = \frac{x}{2} - \left(x + 2\right) \ln{\left(x + 2 \right)} + 1 - \frac{\frac{x^{2}}{2} + 2 x + 2}{x + 2} \]power algebra algebraDifferentiate the remaining terms. Distribute the negative sign. Factor out 1/2 from the numerator.✓ Proved
- \[ = - \left(x + 2\right) \ln{\left(x + 2 \right)} \]algebra algebra algebra algebraRecognize the perfect square trinomial. Simplify the fraction by canceling (x + 2). Distribute the 1/2 into the parentheses. Rearrange terms to group like terms.✓ Proved
- \[ = \left(- x - 2\right) \ln{\left(x + 2 \right)} \]simplifyCombine all terms to find the final simplified derivative.✓ Proved
Answer \( \left(- x - 2\right) \ln{\left(x + 2 \right)} \)
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 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 12 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x + 2 = 0 |
| 13 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x + 2 = 0 |
| 14 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x + 2 = 0 |
| 15 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x + 2 = 0 |
| 16 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x + 2 = 0 |
| 17 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x + 2 = 0 |
| 18 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 19 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 20 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 21 | ✓ 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 12 applies two rules at once (derivative of x and derivative of log), violating the one‑rule‑per‑step rule. Step 14 incorrectly claims to differentiate the remaining terms but leaves them unchanged and mislabels the rule as “power.”qwen3.6:27b-mlx: fail (error) — Step 9 applies the power rule to x**2 but incorrectly leaves Derivative(x, x) in the expression (writing 2*x*Derivative(x, x) instead of just 2*x). This is a fundamental error in applying the power rule, which should resolve the derivative of x**2 to 2*x, not introduce a derivative of x.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-18 — Step 9 applies the power rule to x**2 but incorrectly leaves Derivative(x, x) in the expression (writing 2*x*Derivative(x, x) instead of just 2*x). This is a fundamental error in applying the power rule, which should resolve the derivative of x**2 to 2*x, not introduce a derivative of x.gpt-oss:20b: fail (error) 2026-10-18 — Step 12 applies two rules at once (derivative of x and derivative of log), violating the one‑rule‑per‑step rule. Step 14 incorrectly claims to differentiate the remaining terms but leaves them unchanged and mislabels the rule as “power.”qwen3.6:27b-mlx: fail (error) 2026-10-16 — Step 12 applies the derivative rule for log(x+2) but fails to apply the chain rule to the inner function (x+2), resulting in an incorrect derivative of 1/(x+2) instead of 1/(x+2) * 1. Although the numerical result is coincidentally correct because the derivative of x+2 is 1, the step is logically incomplete as it omits the chain rule application required by the composition.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly applies differentiation rules step-by-step, adhering to the single-change constraint. The algebraic simplifications are valid and clearly labeled.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.