Derivative of \( \displaystyle \sin{\left(x - 3 \right)} \)

Problem 2.35 · medium

Differentiate \( \displaystyle f(x) = \sin{\left(x - 3 \right)} \).

\[ \frac{d}{d x} \sin{\left(x - 3 \right)} \]
derivativeStart with the derivative of the function.✓ Proved
\[ = \cos{\left(x - 3 \right)} \frac{d}{d x} \left(x - 3\right) \]
chainApply the chain rule.✓ Proved
\[ = \left(- \frac{d}{d x} 3 + \frac{d}{d x} x\right) \cos{\left(x - 3 \right)} \]
sumDifferentiate the terms inside the parentheses.✓ Proved
\[ = \cos{\left(x - 3 \right)} \]
constant simplifyThe derivative of a constant is zero. Simplify the final expression.✓ Proved

Answer \( \cos{\left(x - 3 \right)} \)

✓ 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.

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
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 — The solution correctly applies the chain rule, linearity of differentiation (labeled as sum), and basic derivative rules. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.

Every verdict on record (12)

qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies the chain rule, linearity of differentiation (labeled as sum), and basic derivative rules. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
gpt-oss:20b: pass 2026-09-20
qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies the chain rule, sum rule, and constant rule in separate steps. All labels are valid and accurately describe the transformations.
gpt-oss:20b: pass 2026-09-20
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 — The solution correctly applies the chain rule, linearity of differentiation, and basic derivative rules. Each step changes only one aspect of the expression and uses an acceptable label from the fixed vocabulary.
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.