Derivative of \( \displaystyle x^{2} \left(2 \ln{\left(4 x \right)} - 1\right) \)
Problem 2.895 · hard
Differentiate \( \displaystyle f(x) = x^{2} \left(2 \ln{\left(4 x \right)} - 1\right) \).
- \[ \frac{d}{d x} x^{2} \left(2 \ln{\left(4 x \right)} - 1\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = x^{2} \frac{d}{d x} \left(2 \ln{\left(4 x \right)} - 1\right) + \left(2 \ln{\left(4 x \right)} - 1\right) \frac{d}{d x} x^{2} \]productApply the product rule.✓ Proved
- \[ = x^{2} \frac{d}{d x} \left(2 \ln{\left(4 x \right)} - 1\right) + 2 x \left(2 \ln{\left(4 x \right)} - 1\right) \]derivativeDifferentiate the first part of the product.✓ Proved
- \[ = x^{2} \left(- \frac{d}{d x} 1 + \frac{d}{d x} 2 \ln{\left(4 x \right)}\right) + 2 x \left(2 \ln{\left(4 x \right)} - 1\right) \]sumDistribute the derivative over the subtraction.✓ Proved
- \[ = 2 x^{2} \frac{d}{d x} \ln{\left(4 x \right)} + 2 x \left(2 \ln{\left(4 x \right)} - 1\right) \]derivativeDifferentiate the terms inside the parentheses.✓ Proved
- \[ = 2 x \left(2 \ln{\left(4 x \right)} - 1\right) + \frac{x \frac{d}{d x} 4 x}{2} \]chainApply the chain rule to the logarithm.✓ Proved
- \[ = 2 x \left(2 \ln{\left(4 x \right)} - 1\right) + 2 x \]derivative algebra algebra algebraDifferentiate the inner function 4*x. Simplify the fraction. Simplify the expression. Simplify the second term.✓ Proved
- \[ = 4 x \ln{\left(4 x \right)} \]algebra algebra algebraFactor out 2*x. Simplify the expression inside the parentheses. Multiply the terms together.✓ Proved
Answer \( 4 x \ln{\left(4 x \right)} \)
✓ Nihil obstat 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. Reviewers found nothing wrong with the explanation.
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 |
| 13 | ✓ 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 (style) — Step 5 applies a constant‑multiple rule (d/dx[2·log(4x)] = 2·d/dx[log(4x)]) but labels it "derivative". The label should be "constant-multiple".qwen3.6:27b-mlx: pass — The solution correctly applies the product rule, chain rule, and algebraic simplifications. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies the product rule, chain rule, and algebraic simplifications. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: fail (style) 2026-09-26 — Step 5 applies a constant‑multiple rule (d/dx[2·log(4x)] = 2·d/dx[log(4x)]) but labels it "derivative". The label should be "constant-multiple".qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies the product rule, chain rule, and algebraic simplification steps. Each step modifies only one part of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-09-26
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.