Home›Calculus 1›Derivatives of trigonometric functions›Problem 2.214

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

Problem 2.214 · easy Mental math

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

\[ \frac{d}{d x} \sin{\left(x \right)} \]
derivativeStart with the derivative of the function.✓ Proved
\[ = \cos{\left(x \right)} \]
trigThe derivative of sin(x) is cos(x).✓ Proved

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

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
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 (style) — Step 2 is labeled "trig" but the operation performed is the derivative of sin(x), which should be labeled "derivative". The label does not match the rule applied.
deepseek-r1:70b: pass
qwen3.6:27b-mlx: pass — The solution correctly applies the derivative rule to identify the operation and the trig rule to evaluate the derivative of sine. Each step changes only one thing and uses valid labels.

Every verdict on record (15)

qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies the derivative rule to identify the operation and the trig rule to evaluate the derivative of sine. Each step changes only one thing and uses valid labels.
gpt-oss:20b: fail (style) 2026-09-20 — Step 2 is labeled "trig" but the operation performed is the derivative of sin(x), which should be labeled "derivative". The label does not match the rule applied.
qwen3.6:27b-mlx: pass 2026-09-20
gpt-oss:20b: fail (misleading) 2026-09-20 — Step 2 applies the derivative rule, not the trig rule. The label "trig" is incorrect for the differentiation of sin(x).
qwen3.6:27b-mlx: pass 2026-09-19 — The solution correctly applies the derivative rule for sine, and the steps are granular and properly labeled.
deepseek-r1:70b: pass 2026-09-19
gpt-oss:20b: fail (misleading) 2026-09-19 — The second step applies the derivative rule, not a trig rule. Labeling it as "trig" misleads the student about which rule was used.
qwen3.6:27b-mlx: pass 2026-09-19
deepseek-r1:70b: fail (misleading) 2026-09-19 — The second step incorrectly labels the application of the derivative of sin(x) as 'trig' instead of 'derivative'.
gpt-oss:20b: fail (style) 2026-09-19 — Step 2 applies the derivative rule, not a trig simplification; it should be labeled "derivative" instead of "trig".
qwen3.6:27b-mlx: pass 2026-09-18
deepseek-r1:70b: fail (misleading) 2026-09-18 — The second step incorrectly labels the application of the derivative rule as 'trig'.
gpt-oss:20b: fail (style) 2026-09-18 — Step 2 uses the label "trig" but the operation performed is a derivative. The correct label should be "derivative".
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.