Home›Calculus 1›Power rule›Problem 2.643

Derivative of \( \displaystyle \left(x + 1\right)^{4} \)

Problem 2.643 · easy Mental math Video

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 \( 4 \left(x + 1\right)^{3} \)

Top: the graph of f = (x + 1)⁴. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 4(x + 1)³.

✓ 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
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: pass
qwen3.6:27b-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, but the step explicitly introduces Derivative(x+1, x), which is the definition of the chain rule.

Senior review claude-sonnet-5-5, 2026-10-11: fail — The math is correct and every line was proved. Step 4 only drops a factor of 1, so 'constant-multiple' is the wrong label (no derivative rule is applied there) and 'simplify' or 'algebra' would fit. Step 2's 'power' label is acceptable.

qwen3.6:27b-mlx: dismiss — Step 2 applies the power rule to the outer function and adds the inner-derivative factor, so 'power' is a legitimate label, and the contract accepts either label when two fit (chain would also fit).

Every verdict on record (7)

claude-sonnet-5-5: fail (style) 2026-10-11 — The math is correct and every line was proved. Step 4 only drops a factor of 1, so 'constant-multiple' is the wrong label (no derivative rule is applied there) and 'simplify' or 'algebra' would fit. Step 2's 'power' label is acceptable.
qwen3.6:27b-mlx: fail (error) 2026-09-26 — 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, but the step explicitly introduces Derivative(x+1, x), which is the definition of the chain rule.
gpt-oss:20b: pass 2026-09-26
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 2026-09-21
qwen3.6:27b-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.