Derivative of \( \displaystyle \left(x + 1\right)^{3} e^{x + 1} \)
Problem 2.652 · hard Video
Differentiate \( \displaystyle f(x) = \left(x + 1\right)^{3} e^{x + 1} \).
- \[ \frac{d}{d x} \left(x + 1\right)^{3} e^{x + 1} \]Start with the derivative of the function.✓ Proved
- \[ = \left(x + 1\right)^{3} \frac{d}{d x} e^{x + 1} + e^{x + 1} \frac{d}{d x} \left(x + 1\right)^{3} \]productApply the product rule.✓ Proved
- \[ = \left(x + 1\right)^{3} \frac{d}{d x} e^{x + 1} + 3 \left(x + 1\right)^{2} e^{x + 1} \]powerDifferentiate the first part using the power rule and chain rule.✓ Proved
- \[ = \left(x + 1\right)^{3} e^{x + 1} \frac{d}{d x} \left(x + 1\right) + 3 \left(x + 1\right)^{2} e^{x + 1} \]exponentialDifferentiate the exponential part.✓ Proved
- \[ = \left(x + 1\right)^{3} e^{x + 1} + 3 \left(x + 1\right)^{2} e^{x + 1} \]derivative algebraThe derivative of x + 1 is 1. Simplify the expression.✓ Proved
- \[ = \left(x + 1\right)^{2} \left(x + 4\right) e^{x + 1} \]algebra simplifyFactor out the common terms. Final simplification.✓ Proved
Answer \( \left(x + 1\right)^{2} \left(x + 4\right) e^{x + 1} \)
✓ 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 |
| 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 |
| 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: fail (error) — Step 3 applies both the power rule and the chain rule at once, omitting the derivative of the inner function (x+1). The correct step would be 3*(x+1)^2*Derivative(x+1,x)*exp(x+1).qwen3.6:27b-mlx: fail (error) — Step 3 applies both the power rule and the chain rule to differentiate (x+1)^3, violating the one-rule-per-step constraint. Step 4 applies the exponential rule and the chain rule simultaneously to exp(x+1), also violating the constraint.
Senior review claude-sonnet-5-5, 2026-10-11: pass — Every line is correct and was proved by SymPy. The product rule is applied, the two factors are differentiated in separate steps, and the algebra is clean; the labels fall within the allowed vocabulary.
gpt-oss:20b: dismiss — Evaluating the derivative of the simple inner function x+1 (which is 1) is part of the chain-rule step, so 3*(x+1)**2 is correct and SymPy proved the line equal.qwen3.6:27b-mlx: dismiss — Evaluating a derivative of a simple inner function as part of a chain-rule step is one step; step 3 is a single change to one term and step 4 likewise, and the contract allows either label.
Every verdict on record (7)
claude-sonnet-5-5: pass 2026-10-11 — Every line is correct and was proved by SymPy. The product rule is applied, the two factors are differentiated in separate steps, and the algebra is clean; the labels fall within the allowed vocabulary.qwen3.6:27b-mlx: fail (error) 2026-09-26 — Step 3 applies both the power rule and the chain rule to differentiate (x+1)^3, violating the one-rule-per-step constraint. Step 4 applies the exponential rule and the chain rule simultaneously to exp(x+1), also violating the constraint.gpt-oss:20b: fail (error) 2026-09-26 — Step 3 applies both the power rule and the chain rule at once, omitting the derivative of the inner function (x+1). The correct step would be 3*(x+1)^2*Derivative(x+1,x)*exp(x+1).qwen3.6:27b-mlx: fail (style) 2026-09-21 — Step 3 applies both the power rule and the chain rule to differentiate (x + 1)**3, violating the one-rule-per-step constraint. It should be split into a power rule step and a chain rule step.gpt-oss:20b: fail (error) 2026-09-21 — Step 3 applies the power rule to (x+1)^3 but omits the necessary chain rule factor d/dx(x+1)=1. The derivative should be 3*(x+1)^2, not 3*(x+1)^2*exp(x+1).qwen3.6:27b-mlx: fail (error) 2026-09-21 — Step 3 applies both the power rule and the chain rule to differentiate (x + 1)**3, violating the single-rule-per-step constraint. Additionally, the label 'power' does not account for the chain rule application.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.