Derivative of \( \displaystyle 3 x \left(3 x^{2} + 6 x + 4\right) \)
Problem 2.1004 · hard
Differentiate \( \displaystyle f(x) = 3 x \left(3 x^{2} + 6 x + 4\right) \).
- \[ \frac{d}{d x} 3 x \left(3 x^{2} + 6 x + 4\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = 3 \frac{d}{d x} x \left(3 x^{2} + 6 x + 4\right) \]constant-multiplePull out the constant factor 3.✓ Proved
- \[ = 3 x \frac{d}{d x} \left(3 x^{2} + 6 x + 4\right) + 3 \left(3 x^{2} + 6 x + 4\right) \frac{d}{d x} x \]productApply the product rule.✓ Proved
- \[ = 9 x^{2} + 3 x \left(\frac{d}{d x} 4 + \frac{d}{d x} 6 x + \frac{d}{d x} 3 x^{2}\right) + 18 x + 12 \]derivativeDifferentiate each term in the sum.✓ Proved
- \[ = 9 x^{2} + 3 x \left(6 x + 6\right) + 18 x + 12 \]derivative algebraEvaluate the derivatives of the individual terms. Simplify the expression inside the parentheses.✓ Proved
- \[ = 27 x^{2} + 36 x + 12 \]algebra algebra algebraDistribute x into the parentheses. Combine like terms. Distribute the 3 to get the final answer.✓ Proved
Answer \( 3 \left(3 x + 2\right)^{2} \)
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 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 |
| 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: fail (error) — The final answer provided in the task description, 3*(3*x + 2)**2, is mathematically incorrect (it expands to 27*x**2 + 36*x + 12, but the derivative of the original function is 27*x**2 + 36*x + 12? Wait. Let's check. f(x) = 9x^3 + 18x^2 + 12x. f'(x) = 27x^2 + 36x + 12. The stated answer is 3*(9x^2 + 12x + 4) = 27x^2 + 36x + 12. Okay, the stated answer IS correct. The steps arrive at 27*x**2 + 36*x + 12. So the math is correct. Why did I think it was wrong? Let me re-read the contract. 'A computer algebra system has already confirmed that consecutive lines are algebraically equal'. So I don't need to check the algebra. I need to check the rules. Step 4 applies 'derivative' to a sum. The rule 'sum' is in the vocabulary. Step 4 does Derivative(A+B+C) -> Derivative(A)+Derivative(B)+Derivative(C). This is the sum rule. The label is 'derivative'. Is 'derivative' a valid label for applying the sum rule? The contract says: 'derivative' means unfolding d/dx on a known form (the derivative of x, of a constant, of sin, of exp). It does NOT say 'derivative' covers the sum rule. The sum rule is 'sum'. Therefore, labeling the application of the sum rule as 'derivative' is a defect because 'derivative' is reserved for basic derivatives, and 'sum' is the specific label for linearity over addition. Also, Step 4 applies the sum rule AND evaluates Derivative(x,x)=1. That is two rules: sum and derivative. This violates the 'one thing per step' rule. Step 4 is a multi-rule step labeled incorrectly.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The final answer provided in the task description, 3*(3*x + 2)**2, is mathematically incorrect (it expands to 27*x**2 + 36*x + 12, but the derivative of the original function is 27*x**2 + 36*x + 12? Wait. Let's check. f(x) = 9x^3 + 18x^2 + 12x. f'(x) = 27x^2 + 36x + 12. The stated answer is 3*(9x^2 + 12x + 4) = 27x^2 + 36x + 12. Okay, the stated answer IS correct. The steps arrive at 27*x**2 + 36*x + 12. So the math is correct. Why did I think it was wrong? Let me re-read the contract. 'A computer algebra system has already confirmed that consecutive lines are algebraically equal'. So I don't need to check the algebra. I need to check the rules. Step 4 applies 'derivative' to a sum. The rule 'sum' is in the vocabulary. Step 4 does Derivative(A+B+C) -> Derivative(A)+Derivative(B)+Derivative(C). This is the sum rule. The label is 'derivative'. Is 'derivative' a valid label for applying the sum rule? The contract says: 'derivative' means unfolding d/dx on a known form (the derivative of x, of a constant, of sin, of exp). It does NOT say 'derivative' covers the sum rule. The sum rule is 'sum'. Therefore, labeling the application of the sum rule as 'derivative' is a defect because 'derivative' is reserved for basic derivatives, and 'sum' is the specific label for linearity over addition. Also, Step 4 applies the sum rule AND evaluates Derivative(x,x)=1. That is two rules: sum and derivative. This violates the 'one thing per step' rule. Step 4 is a multi-rule step labeled incorrectly.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: pass 2026-09-27gpt-oss:20b: pass 2026-09-27
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.