Derivative of \( \displaystyle 5 x^{3} \)

Problem 2.37 · easy Video

Differentiate \( \displaystyle f(x) = 5 x^{3} \).

\[ \frac{d}{d x} 5 x^{3} \]
derivativeStart with the derivative of the function.✓ Proved
\[ = 5 \frac{d}{d x} x^{3} \]
constant-multiplePull the constant factor out of the derivative.✓ Proved
\[ = 15 x^{2} \]
power simplifyApply the power rule to x**3. Multiply the constants to get the final result.✓ Proved

Answer \( 15 x^{2} \)

Top: the graph of f = 5x³. A tangent line slides along it, and its slope is plotted below as it goes, tracing f' = 15x².

✓ 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
deepseek-r1:70b: pass
qwen3.6:27b-mlx: pass

Senior review claude-sonnet-5-5, 2026-10-03: pass — All steps are correct and use valid rule names: constant-multiple, then power, then simplify for 5*3*x**2 = 15*x**2. SymPy proved every line and the final answer. The only panel item is a note that a model both passed and objected, with no specific objection to rule on.

Every verdict on record (13)

claude-sonnet-5-5: pass 2026-10-03 — All steps are correct and use valid rule names: constant-multiple, then power, then simplify for 5*3*x**2 = 15*x**2. SymPy proved every line and the final answer. The only panel item is a note that a model both passed and objected, with no specific objection to rule on.
qwen3.6:27b-mlx: pass 2026-09-20
gpt-oss:20b: pass 2026-09-20
qwen3.6:27b-mlx: pass 2026-09-20
gpt-oss:20b: fail (error) 2026-09-20 — Step 1 applies two rules at once: it takes the derivative of the product 5*x**3 and immediately pulls out the constant 5. This violates the rule that each step must change only one thing. The correct first step would be to apply the constant‑multiple rule, yielding 5*Derivative(x**3,x).
qwen3.6:27b-mlx: pass 2026-09-19
deepseek-r1:70b: pass 2026-09-19
gpt-oss:20b: pass 2026-09-19
qwen3.6:27b-mlx: pass 2026-09-19
deepseek-r1:70b: pass 2026-09-19
gpt-oss:20b: pass 2026-09-19
gpt-oss:20b: pass 2026-09-17
deepseek-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.