Derivative of \( \displaystyle \left(x + 1\right)^{4} \)
Problem 2.643 · easy Mental math
Differentiate \( \displaystyle f(x) = \left(x + 1\right)^{4} \).
- \[ \frac{d}{d x} \left(x + 1\right)^{4} \]Start with the derivative of the function.✓ Proved
- \[ = 4 \left(x + 1\right)^{3} \frac{d}{d x} \left(x + 1\right) \]powerApply the power rule.✓ Proved
- \[ = 4 \left(x + 1\right)^{3} \]derivative constant-multipleDifferentiate the inner function x + 1. Simplify the expression.✓ Proved
Answer \( 4 \left(x + 1\right)^{3} \)
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 |
| 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) — Step 2 is labeled 'power' but applies the chain rule (differentiating the outer function and multiplying by the derivative of the inner function). The power rule alone would yield 4*(x+1)^3 without the extra derivative term, or requires the chain rule label if the inner derivative is kept. Applying two rules (power and chain) in one step violates the single-rule constraint.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-21 — Step 2 is labeled 'power' but applies the chain rule (differentiating the outer function and multiplying by the derivative of the inner function). The power rule alone would yield 4*(x+1)^3 without the extra derivative term, or requires the chain rule label if the inner derivative is kept. Applying two rules (power and chain) in one step violates the single-rule constraint.gpt-oss:20b: pass 2026-09-21qwen3.6:27b-mlx: fail (error) 2026-09-21 — Step 2 is labeled 'power', but it applies the chain rule (differentiating the outer power function while keeping the inner derivative unevaluated). The power rule alone would not produce the Derivative(x + 1, x) term; this is a mislabeling of the chain rule.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.