Derivative of \( \displaystyle 20 x \left(4 x^{2} + 6 x + 3\right) \)
Problem 2.114 · hard
Differentiate \( \displaystyle f(x) = 20 x \left(4 x^{2} + 6 x + 3\right) \).
- \[ \frac{d}{d x} 20 x \left(4 x^{2} + 6 x + 3\right) \]derivativeStart with the derivative of the function.✓ Proved
- \[ = 20 \frac{d}{d x} x \left(4 x^{2} + 6 x + 3\right) \]constant-multiplePull out the constant factor 20.✓ Proved
- \[ = 20 x \frac{d}{d x} \left(4 x^{2} + 6 x + 3\right) + 20 \left(4 x^{2} + 6 x + 3\right) \frac{d}{d x} x \]productApply the product rule.✓ Proved
- \[ = 80 x^{2} + 20 x \left(\frac{d}{d x} 3 + \frac{d}{d x} 6 x + \frac{d}{d x} 4 x^{2}\right) + 120 x + 60 \]derivativeDifferentiate each term in the sum.✓ Proved
- \[ = 80 x^{2} + 20 x \left(8 x + 6\right) + 120 x + 60 \]derivative algebraEvaluate the derivatives of the individual terms. Simplify the expression inside the parentheses.✓ Proved
- \[ = 240 x^{2} + 240 x + 60 \]algebra algebra algebraDistribute x into the parentheses. Combine like terms. Distribute the 20 to get the final simplified form.✓ Proved
Answer \( 60 \left(2 x + 1\right)^{2} \)
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: passdeepseek-r1:70b: passqwen3.6:27b-mlx: pass — The solution correctly applies differentiation rules and algebraic simplifications. Each step changes only one aspect of the expression and uses valid labels from the vocabulary.
Every verdict on record (15)
qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies differentiation rules and algebraic simplifications. Each step changes only one aspect of the expression and uses valid labels from the vocabulary.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies differentiation rules and algebraic simplifications step-by-step. The final result matches the stated answer.gpt-oss:20b: pass 2026-09-20qwen3.6:27b-mlx: pass 2026-09-19deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: fail (error) 2026-09-19 — The final answer provided in the task description, 60*(2*x + 1)**2, is mathematically incorrect (it expands to 240*x**2 + 240*x + 60, but the derivative of the original function is 240*x**2 + 240*x + 60... wait, let me recheck. f(x) = 80x^3 + 120x^2 + 60x. f'(x) = 240x^2 + 240x + 60. The stated answer 60*(2x+1)^2 = 60*(4x^2+4x+1) = 240x^2 + 240x + 60. The stated answer IS correct. However, the steps end at step 9 with 240*x**2 + 240*x + 60. The task asks to differentiate, and the 'Stated answer' is a specific form. The steps do not reach the stated answer form. But more importantly, look at Step 4. It applies 'derivative' to split the sum inside the derivative. This is valid. Step 5 evaluates them. Step 6 simplifies. Step 7 distributes. Step 8 combines. Step 9 distributes 20. The steps are algebraically correct and lead to the expanded form. The 'Stated answer' is just a different form of the same polynomial. Is there a defect in the steps? Step 4 label 'derivative' is used for linearity of differentiation (sum rule). The vocabulary allows 'sum' or 'derivative'. 'derivative' is acceptable for unfolding d/dx on a sum if interpreted broadly, but usually 'sum' is the rule for d(u+v)=du+dv. However, the prompt says 'derivative' means unfolding d/dx on a known form. Differentiating a sum is a known form. Let's look closer. Step 4: `Derivative(4*x**2 + 6*x + 3, x)` becomes `Derivative(4*x**2, x) + ...`. This is the sum rule. The label is 'derivative'. Is 'sum' a better label? The vocabulary has 'sum'. The prompt says 'sum' is for steps that apply the sum rule. Step 4 applies the sum rule. Labeling it 'derivative' is ambiguous but perhaps acceptable if 'derivative' covers 'apply derivative operator'. But wait, Step 4 also evaluates `Derivative(x, x)` to `1`. This is TWO things: applying the sum rule to the second term AND evaluating the derivative of x. Step 3 was `20*(Derivative(x, x)*(...) + x*Derivative(...))`. Step 4 is `20*(1*(...) + x*(Derivative(4*x**2, x) + ...))`. It changed `Derivative(x, x)` to `1` AND split `Derivative(4*x**2 + 6*x + 3, x)` into a sum. This is two rules: evaluating a basic derivative (derivative) and linearity (sum). This violates the 'one rule per step' constraint.deepseek-r1:70b: pass 2026-09-19gpt-oss:20b: pass 2026-09-19qwen3.6:27b-mlx: pass 2026-09-18 — The solution correctly applies differentiation rules step-by-step and simplifies the result algebraically. The final answer matches the stated answer.deepseek-r1:70b: pass 2026-09-18gpt-oss:20b: pass 2026-09-18gpt-oss:20b: pass 2026-09-17deepseek-r1:70b: pass 2026-09-17
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.