Home›Calculus 3›Implicit differentiation with partial derivatives›Problem 10.411

Implicit differentiation with partial derivatives

Problem 10.411 · easy

Use partial derivatives to find \( \displaystyle \frac{dy}{dx} \) at \( \displaystyle (0, -1) \) for the curve \( \displaystyle x y + \sin{\left(x \right)} + \cos{\left(y \right)} = \cos{\left(1 \right)} \).

\[ \left[\begin{matrix}\frac{\partial}{\partial x} \left(x y + \sin{\left(x \right)} + \cos{\left(y \right)}\right)\\\frac{\partial}{\partial y} \left(x y + \sin{\left(x \right)} + \cos{\left(y \right)}\right)\end{matrix}\right] = \left[\begin{matrix}y + \cos{\left(x \right)}\\x - \sin{\left(y \right)}\end{matrix}\right] \]
F_x and F_y.✓ Proved
\[ 0 \]
dy/dx = −F_x/F_y at the point.✓ Proved

Answer \( 0 \)

✓ 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	simplify(lhs - rhs) reduced to 0
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	solved the equation numerically near the point and differenced

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (style) — [domain objection, downgraded to style] The solution fails to substitute the point (0, -1) into the partial derivatives before applying the formula dy/dx = -F_x/F_y. It incorrectly asserts that the result is 0 without showing the calculation F_x(0,-1) = 1 and F_y(0,-1) = 0, which actually indicates a vertical tangent (undefined derivative), not 0.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-07
qwen3.6:27b-mlx: fail (style) 2026-10-07 — [domain objection, downgraded to style] The solution fails to substitute the point (0, -1) into the partial derivatives before applying the formula dy/dx = -F_x/F_y. It incorrectly asserts that the result is 0 without showing the calculation F_x(0,-1) = 1 and F_y(0,-1) = 0, which actually indicates a vertical tangent (undefined derivative), not 0.
gpt-oss:20b: pass 2026-10-07
qwen3.6:27b-mlx: fail (style) 2026-10-07 — [domain objection, downgraded to style] The solution fails to evaluate the partial derivatives at the point (0, -1) before applying the formula dy/dx = -F_x/F_y. Specifically, F_y(0, -1) = 0 - sin(-1) = sin(1) != 0, so the derivative is defined, but the solution incorrectly states 0=0 without showing the substitution or the final calculation.

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/implicit_partials, checked 2026-10-07 with SymPy 1.14.0.