Derivative of \( \displaystyle x^{2} \left(6 \ln{\left(4 x \right)} - 3\right) \)
Problem 2.1116 · hard Video
Differentiate \( \displaystyle f(x) = x^{2} \left(6 \ln{\left(4 x \right)} - 3\right) \).
- \[ \frac{d}{d x} x^{2} \left(6 \ln{\left(4 x \right)} - 3\right) \]Start with the derivative of the function.✓ Proved
- \[ = x^{2} \frac{d}{d x} \left(6 \ln{\left(4 x \right)} - 3\right) + \left(6 \ln{\left(4 x \right)} - 3\right) \frac{d}{d x} x^{2} \]productApply the product rule.✓ Proved
- \[ = x^{2} \frac{d}{d x} \left(6 \ln{\left(4 x \right)} - 3\right) + 2 x \left(6 \ln{\left(4 x \right)} - 3\right) \]derivativeDifferentiate the first part of the product.✓ Proved
- \[ = x^{2} \left(- \frac{d}{d x} 3 + \frac{d}{d x} 6 \ln{\left(4 x \right)}\right) + 2 x \left(6 \ln{\left(4 x \right)} - 3\right) \]sumDistribute the derivative over the subtraction.✓ Proved
- \[ = 6 x^{2} \frac{d}{d x} \ln{\left(4 x \right)} + 2 x \left(6 \ln{\left(4 x \right)} - 3\right) \]derivative constantDifferentiate the terms inside the parentheses. The derivative of the constant 3 is 0.✓ Proved
- \[ = 2 x \left(6 \ln{\left(4 x \right)} - 3\right) + \frac{3 x \frac{d}{d x} 4 x}{2} \]chainApply the chain rule to log(4*x).✓ Proved
- \[ = 2 x \left(6 \ln{\left(4 x \right)} - 3\right) + 6 x \]derivative algebra algebraDifferentiate 4*x. Simplify the expression inside the second term. Simplify the product of x**2 and 6/x.✓ Proved
- \[ = 12 x \ln{\left(4 x \right)} \]algebra simplifyDistribute 2*x into the parentheses. Combine like terms.✓ Proved
Answer \( 12 x \ln{\left(4 x \right)} \)
✓ 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 |
| 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 correspond to the rules applied, and the final result is correct.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies the product rule, chain rule, and algebraic simplifications in distinct steps. All labels correspond to the rules applied, and the final result is correct.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: pass 2026-09-28gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly applies the product rule, linearity of differentiation, and chain rule in separate, well-labeled steps. The algebraic simplifications are accurate and lead to the correct final result.gpt-oss:20b: pass 2026-09-28
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 2026-10-03 with SymPy 1.14.0.