Derivative of \( \displaystyle 8 x^{3} e^{2 x} \)
Problem 2.645 · hard
Differentiate \( \displaystyle f(x) = 8 x^{3} e^{2 x} \).
- \[ \frac{d}{d x} 8 x^{3} e^{2 x} \]derivativeStart with the derivative of the function.✓ Proved
- \[ = 8 \frac{d}{d x} x^{3} e^{2 x} \]constant-multiplePull out the constant factor 8.✓ Proved
- \[ = 8 x^{3} \frac{d}{d x} e^{2 x} + 8 e^{2 x} \frac{d}{d x} x^{3} \]productApply the product rule.✓ Proved
- \[ = 8 x^{3} \frac{d}{d x} e^{2 x} + 24 x^{2} e^{2 x} \]powerDifferentiate the first part of the product.✓ Proved
- \[ = 16 x^{3} e^{2 x} + 24 x^{2} e^{2 x} \]exponential algebraDifferentiate the exponential part using the chain rule. Rearrange the terms.✓ Proved
- \[ = 8 x^{2} \left(2 x + 3\right) e^{2 x} \]simplifyFactor out the common terms.✓ Proved
Answer \( x^{2} \left(16 x + 24\right) e^{2 x} \)
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 |
| 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 (style) — Step 5 is labeled 'exponential', but the note explicitly states 'using the chain rule'. While the label 'exponential' is acceptable for the derivative of exp, the step applies the chain rule to the inner function 2x, which is a distinct rule application. More critically, the label 'exponential' typically refers to d/dx(e^x) = e^x, whereas this step computes d/dx(e^(2x)) = 2e^(2x), which is a composite application. However, the primary defect is that Step 5 applies the chain rule (differentiating the inner function 2x) but labels it 'exponential'. The vocabulary includes 'chain', so 'chain' would be the more precise label for the differentiation of the composite function, or the step should be split. But looking closer at the contract: 'exponential' is in the list. Is it wrong? The step computes Derivative(exp(2*x), x). This is a chain rule application. The label 'exponential' is ambiguous. Let's look at Step 4. Step 4 computes Derivative(x**3, x) -> 3x**2. This is 'power'. Correct. Step 5 computes Derivative(exp(2*x), x) -> 2*exp(2*x). This requires the chain rule. The label is 'exponential'. The note says 'using the chain rule'. The label 'exponential' usually implies the base rule d/dx e^u = e^u u'. If the label 'chain' exists, it should be used for the chain rule part. However, often 'exponential' covers the whole derivative of an exponential function. Let's re-read the constraints. 'A step that applies two rules at once is a defect.' Step 5 applies the chain rule. It does not apply two rules. It applies one rule (chain rule to exp(2x)). The label 'exponential' might be considered a misnomer if 'chain' is available. But wait, Step 5 is `8*(3*x**2*exp(2*x) + x**3*Derivative(exp(2*x), x))` -> `8*(3*x**2*exp(2*x) + x**3*2*exp(2*x))`. The change is `Derivative(exp(2*x), x)` to `2*exp(2*x)`. This is the derivative of a composite function. The rule is Chain Rule. The label is 'exponential'. This is a labeling defect. The label should be 'chain'. The note says 'using the chain rule', which contradicts the label 'exponential' if we assume labels must be precise. Actually, 'exponential' is a valid label for the derivative of an exponential function, but since 'chain' is in the vocabulary and the step *is* a chain rule step, 'chain' is the correct label. Using 'exponential' is misleading because it hides the chain rule aspect which is the non-trivial part here. Furthermore, Step 5 note says 'using the chain rule' but label is 'exponential'. This inconsistency is a defect.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (style) 2026-09-21 — Step 5 is labeled 'exponential', but the note explicitly states 'using the chain rule'. While the label 'exponential' is acceptable for the derivative of exp, the step applies the chain rule to the inner function 2x, which is a distinct rule application. More critically, the label 'exponential' typically refers to d/dx(e^x) = e^x, whereas this step computes d/dx(e^(2x)) = 2e^(2x), which is a composite application. However, the primary defect is that Step 5 applies the chain rule (differentiating the inner function 2x) but labels it 'exponential'. The vocabulary includes 'chain', so 'chain' would be the more precise label for the differentiation of the composite function, or the step should be split. But looking closer at the contract: 'exponential' is in the list. Is it wrong? The step computes Derivative(exp(2*x), x). This is a chain rule application. The label 'exponential' is ambiguous. Let's look at Step 4. Step 4 computes Derivative(x**3, x) -> 3x**2. This is 'power'. Correct. Step 5 computes Derivative(exp(2*x), x) -> 2*exp(2*x). This requires the chain rule. The label is 'exponential'. The note says 'using the chain rule'. The label 'exponential' usually implies the base rule d/dx e^u = e^u u'. If the label 'chain' exists, it should be used for the chain rule part. However, often 'exponential' covers the whole derivative of an exponential function. Let's re-read the constraints. 'A step that applies two rules at once is a defect.' Step 5 applies the chain rule. It does not apply two rules. It applies one rule (chain rule to exp(2x)). The label 'exponential' might be considered a misnomer if 'chain' is available. But wait, Step 5 is `8*(3*x**2*exp(2*x) + x**3*Derivative(exp(2*x), x))` -> `8*(3*x**2*exp(2*x) + x**3*2*exp(2*x))`. The change is `Derivative(exp(2*x), x)` to `2*exp(2*x)`. This is the derivative of a composite function. The rule is Chain Rule. The label is 'exponential'. This is a labeling defect. The label should be 'chain'. The note says 'using the chain rule', which contradicts the label 'exponential' if we assume labels must be precise. Actually, 'exponential' is a valid label for the derivative of an exponential function, but since 'chain' is in the vocabulary and the step *is* a chain rule step, 'chain' is the correct label. Using 'exponential' is misleading because it hides the chain rule aspect which is the non-trivial part here. Furthermore, Step 5 note says 'using the chain rule' but label is 'exponential'. This inconsistency is a defect.gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: fail (misleading) 2026-09-21 — Step 5 is labeled 'exponential', but the differentiation of exp(2*x) requires the chain rule (derivative of the inner function 2*x). The label 'exponential' typically refers to the basic rule d/dx exp(x) = exp(x) and does not account for the inner derivative, making the label inaccurate for this composite function step.gpt-oss:20b: pass 2026-09-21
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.