Derivative of \( \displaystyle x^{2} \left(2 \ln{\left(4 x \right)} - 1\right) \)
Problem 2.895 · hard Video
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. 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 |
| 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: passqwen3.6:27b-mlx: pass — The solution correctly applies the product rule, chain rule, and algebraic simplifications in distinct steps. All labels accurately reflect the operations performed.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly applies the product rule, chain rule, and algebraic simplifications in distinct steps. All labels accurately reflect the operations performed.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-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 2026-09-27 with SymPy 1.14.0.