Home›Calculus 1›Derivatives of trigonometric functions›Problem 2.101

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

Problem 2.101 · easy Mental math

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

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

Answer \( - \sin{\left(x \right)} \)

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 (misleading) — Step 2 incorrectly labels the derivative of cos(x) as a "trig" rule; the correct rule is "derivative". Moreover, step 2 is redundant—step 1 already yields the result. This could mislead a student into thinking a separate trig rule is needed for differentiation.
deepseek-r1:70b: fail (misleading) — The step labels the application of the derivative rule as 'trig', which is misleading. It should be labeled 'derivative' to accurately reflect the operation performed.
qwen3.6:27b-mlx: pass

Every verdict on record (15)

qwen3.6:27b-mlx: pass 2026-09-20
gpt-oss:20b: fail (misleading) 2026-09-20 — Step 2 incorrectly labels the derivative of cos(x) as a "trig" rule; the correct rule is "derivative". Moreover, step 2 is redundant—step 1 already yields the result. This could mislead a student into thinking a separate trig rule is needed for differentiation.
qwen3.6:27b-mlx: pass 2026-09-20 — The solution correctly applies the derivative rule for cosine. Each step changes only one thing and uses valid labels from the fixed vocabulary.
gpt-oss:20b: fail (style) 2026-09-20 — Step 2 is labeled "trig" but it is applying the derivative rule for cos(x); the correct label should be "derivative".
qwen3.6:27b-mlx: pass 2026-09-19 — The solution correctly applies the derivative rule for cosine in a single step. The label 'trig' is appropriate for applying a standard trigonometric derivative identity.
deepseek-r1:70b: fail (misleading) 2026-09-19 — The step labels the application of the derivative rule as 'trig', which is misleading. It should be labeled 'derivative' to accurately reflect the operation performed.
gpt-oss:20b: pass 2026-09-19
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 rule as 'trig', which could mislead a student into believing trigonometric identities were used instead of differentiation.
gpt-oss:20b: pass 2026-09-19
qwen3.6:27b-mlx: pass 2026-09-18
deepseek-r1:70b: pass 2026-09-18
gpt-oss:20b: fail (style) 2026-09-18 — Step 2 applies the derivative rule, not a trig‑specific rule. The correct label should be "derivative" (or "simplify" if treating it as a presentation step).
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.