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Tangent lines

Problem 2.779 · medium Video

Find an equation of the tangent line to \( \displaystyle y = - x - \sin{\left(x \right)} \) at \( \displaystyle x = 0 \).

The tangent line passes through the point (a, f(a)) and has slope f'(a).
\[ \left. - x - \sin{\left(x \right)} \right|_{\substack{ x=0 }} = 0 \]
The point of tangency.✓ Proved
\[ \frac{d}{d x} \left(- x - \sin{\left(x \right)}\right) = - \cos{\left(x \right)} - 1 \]
Differentiate.✓ Proved
\[ \left. - \cos{\left(x \right)} - 1 \right|_{\substack{ x=0 }} = -2 \]
The slope at the point.✓ Proved
\[ - 2 x \]
Point-slope form, then simplify.

Answer \( y = - 2 x \)

The graph of y = -x - sin(x) near x = 0. A secant line through the point (0, 0) and a second point on the curve turns into the tangent line as the second point slides in; its slope settles at -2, giving y = -2x.

Lines: 3 proved, 2 not checked. 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	Not checked	claude-sonnet-5-5 script, run by sympy 1.14.0	claude-sonnet-5-5 script, run by sympy 1.14.0: script failed: SyntaxError: invalid syntax (`checks/2.779/line1-e8c9cfb6.py`)
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
3	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
4	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
5	Not checked	—	both sides are the same expression: the line states a value, and the working that produced it is not on the line
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	the line meets the curve at x = a, and its slope matches a central difference quotient of f there

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (error) — The solution fails to explicitly state the point-slope formula or substitute the point (0,0) and slope -2 into it. Line 5 jumps directly to the simplified equation without showing the intermediate step y - 0 = -2(x - 0), making the derivation incomplete.

Senior review claude-sonnet-5-5, 2026-10-12: pass — The point (0,0), derivative -cos(x)-1, and slope -2 are all correct, so the tangent line y = -2x is correct. Line 5 combines point-slope substitution with simplification, but that is a minor presentational choice and not a defect.

qwen3.6:27b-mlx: dismiss — Line 5 is labeled 'Point-slope form, then simplify', which names the rule used. The result y = -2x is correct, and the point and slope are established in lines 2 and 4. Skipping the intermediate y - 0 = -2(x - 0) is at most a mild style issue and not a false statement.

Every verdict on record (7)

claude-sonnet-5-5: pass 2026-10-12 — The point (0,0), derivative -cos(x)-1, and slope -2 are all correct, so the tangent line y = -2x is correct. Line 5 combines point-slope substitution with simplification, but that is a minor presentational choice and not a defect.
qwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution fails to explicitly state the point-slope formula or substitute the point (0,0) and slope -2 into it. Line 5 jumps directly to the simplified equation without showing the intermediate step y - 0 = -2(x - 0), making the derivation incomplete.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The solution fails to explicitly state the point-slope formula or substitute the point (0,0) and slope -2 into it. Line 5 jumps directly to the simplified equation without showing the intermediate step y - 0 = -2(x - 0), which is necessary to justify the result.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The solution fails to explicitly state the point-slope formula or substitute the point (0,0) and slope -2 into it. Line 5 jumps directly to the final expression '-2*x' without showing the intermediate step 'y - 0 = -2(x - 0)', making the derivation incomplete and potentially confusing for a student.
gpt-oss:20b: pass 2026-09-26

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 generator:structured/tangent_line, checked 2026-10-13 with SymPy 1.14.0.