Derivative of \( \displaystyle - \frac{3 x^{2}}{4} + \ln{\left(3^{\frac{3 x^{2}}{2}} x^{\frac{3 x^{2}}{2}} \right)} \)
Problem 2.1748 · hard
Differentiate \( \displaystyle f(x) = - \frac{3 x^{2}}{4} + \ln{\left(3^{\frac{3 x^{2}}{2}} x^{\frac{3 x^{2}}{2}} \right)} \).
- \[ \frac{d}{d x} \left(- \frac{3 x^{2}}{4} + \ln{\left(3^{\frac{3 x^{2}}{2}} x^{\frac{3 x^{2}}{2}} \right)}\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = \frac{d}{d x} \left(- \frac{3 x^{2}}{4}\right) + \frac{d}{d x} \ln{\left(3^{\frac{3 x^{2}}{2}} x^{\frac{3 x^{2}}{2}} \right)} \]sumApply the sum rule.✓ Proved
- \[ = \frac{d}{d x} \left(- \frac{3 x^{2}}{4}\right) + \frac{d}{d x} \ln{\left(e^{\frac{3 x^{2} \ln{\left(x \right)}}{2} + \frac{3 x^{2} \ln{\left(3 \right)}}{2}} \right)} \]rewriteRewrite the logarithm of a product with exponents using the exponential and logarithm identity.✓ Proved
- \[ = \frac{d}{d x} \left(- \frac{3 x^{2}}{4}\right) + \frac{d}{d x} \left(\frac{3 x^{2} \ln{\left(x \right)}}{2} + \frac{3 x^{2} \ln{\left(3 \right)}}{2}\right) \]logarithmicApply the derivative rule for the natural logarithm.✓ Proved
- \[ = \frac{d}{d x} \left(- \frac{3 x^{2}}{4}\right) + \frac{d}{d x} \frac{3 x^{2} \ln{\left(3 \right)}}{2} + \frac{d}{d x} \frac{3 x^{2} \ln{\left(x \right)}}{2} \]sumApply the sum rule to the expanded expression.✓ Proved
- \[ = \frac{d}{d x} \left(- \frac{3 x^{2}}{4}\right) + \ln{\left(3 \right)} \frac{d}{d x} \frac{3 x^{2}}{2} + \frac{d}{d x} \frac{3 x^{2} \ln{\left(x \right)}}{2} \]constant-multipleFactor out the constant log(3).✓ Proved
- \[ = \frac{3 x^{2} \frac{d}{d x} \ln{\left(x \right)}}{2} + \ln{\left(x \right)} \frac{d}{d x} \frac{3 x^{2}}{2} + \frac{d}{d x} \left(- \frac{3 x^{2}}{4}\right) + \ln{\left(3 \right)} \frac{d}{d x} \frac{3 x^{2}}{2} \]productApply the product rule to the term containing x.✓ Proved
- \[ = 3 x \ln{\left(x \right)} + \frac{3 x}{2} + 3 x \ln{\left(3 \right)} + \frac{d}{d x} \left(- \frac{3 x^{2}}{4}\right) \]powerDifferentiate the power functions and the logarithm.✓ Proved
- \[ = 3 x \ln{\left(x \right)} + 3 x \ln{\left(3 \right)} \]derivative algebraEvaluate the derivatives of the individual terms. Simplify the expression by combining like terms.✓ Proved
- \[ = 3 x \left(\ln{\left(x \right)} + \ln{\left(3 \right)}\right) \]algebraFactor out the common term 3*x.✓ Proved
- \[ = 3 x \ln{\left(3 x \right)} \]simplifyUse the logarithm product rule to simplify the expression.✓ Proved
Answer \( 3 x \ln{\left(3 x \right)} \)
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.
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 |
| 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 8 applies two differentiation rules at once (power for (3*x**2/2) and logarithmic for log(x)), violating the one‑rule‑per‑step rule.qwen3.6:27b-mlx: pass — The solution correctly applies differentiation rules and algebraic simplifications in a step-by-step manner. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-07 — Step 8 applies two differentiation rules at once (power for (3*x**2/2) and logarithmic for log(x)), violating the one‑rule‑per‑step rule.qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly applies differentiation rules and algebraic simplifications in a step-by-step manner. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: fail (error) 2026-10-07 — Step 4 incorrectly applies the logarithmic derivative rule: the derivative of log(u) is u'/u, not simply u'. The subsequent steps propagate this mistake, leading to an incorrect final result.qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly applies differentiation rules and algebraic simplifications in a step-by-step manner. Each step changes only one aspect of the expression and uses appropriate labels from the fixed vocabulary.
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 2026-10-07 with SymPy 1.14.0.