Implicit differentiation
Problem 2.1475 · hard
The curve \( \displaystyle - 4 x + 2 y + \sin{\left(x y \right)} = 6 - \sin{\left(1 \right)} \) passes through \( \displaystyle (-1, 1) \). Find \( \displaystyle \dfrac{dy}{dx} \) by implicit differentiation, and its value at that point.
- \[ 6 - \sin{\left(1 \right)} \]The point is on the curve.✓ Proved
- Differentiate both sides with respect to x, treating y as a function of x, then solve for dy/dx.
- \[ \frac{d}{d x} \left(- 4 x + 2 Y{\left(x \right)} + \sin{\left(x Y{\left(x \right)} \right)}\right) = \left(x \cos{\left(x Y{\left(x \right)} \right)} + 2\right) \frac{d}{d x} Y{\left(x \right)} + Y{\left(x \right)} \cos{\left(x Y{\left(x \right)} \right)} - 4 \]Every y term picks up a factor dy/dx.✓ Proved
- \[ \frac{- y \cos{\left(x y \right)} + 4}{x \cos{\left(x y \right)} + 2} \]Solve for dy/dx: minus F_x over F_y.✓ Proved
- \[ \frac{4 - \cos{\left(1 \right)}}{2 - \cos{\left(1 \right)}} \]At the point.✓ Proved
Answer \( \frac{dy}{dx} = \frac{- y \cos{\left(x y \right)} + 4}{x \cos{\left(x y \right)} + 2}, \quad \left.\frac{dy}{dx}\right|_{(-1,1)} = \frac{4 - \cos{\left(1 \right)}}{2 - \cos{\left(1 \right)}} \)
Lines: 4 proved, 1 not checked. 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 | simplify(lhs - rhs) reduced to 0 |
| 2 | Not checked | — | a sentence; read, not computed |
| 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 | ✓ 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 | sympy.idiff differentiates the curve on its own and agrees |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: fail (error) — The comment on line 3 claims that 'Every y term picks up a factor dy/dx', which is mathematically false for the term sin(xy) where x is also present. The derivative of sin(xy) involves both y*cos(xy)*y' and x*cos(xy), not just a factor of y'. This incorrect reasoning obscures the application of the product rule.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-03 — The comment on line 3 claims that 'Every y term picks up a factor dy/dx', which is mathematically false for the term sin(xy) where x is also present. The derivative of sin(xy) involves both y*cos(xy)*y' and x*cos(xy), not just a factor of y'. This incorrect reasoning obscures the application of the product rule.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies implicit differentiation, handles the chain rule for the sin(xy) term, and evaluates the derivative at the specified point. The intermediate steps and final result are accurate.gpt-oss:20b: pass 2026-10-03
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_differentiation, checked 2026-10-03 with SymPy 1.14.0.