Implicit differentiation
Problem 2.810 · medium
The curve \( \displaystyle x^{3} + 2 x y + y^{3} = 1 \) passes through \( \displaystyle (1, 0) \). Find \( \displaystyle \dfrac{dy}{dx} \) by implicit differentiation, and its value at that point.
- \[ 1 \]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.
- \[ \frac{d}{d x} \left(x^{3} + 2 x Y{\left(x \right)} + Y^{3}{\left(x \right)}\right) = 3 x^{2} + \left(2 x + 3 Y^{2}{\left(x \right)}\right) \frac{d}{d x} Y{\left(x \right)} + 2 Y{\left(x \right)} \]Every y term picks up a factor dy/dx.✓ Proved
- \[ \frac{- 3 x^{2} - 2 y}{2 x + 3 y^{2}} \]Solve for dy/dx: minus F_x over F_y.✓ Proved
- \[ - \frac{3}{2} \]At the point.✓ Proved
Answer \( \frac{dy}{dx} = \frac{- 3 x^{2} - 2 y}{2 x + 3 y^{2}}, \quad \left.\frac{dy}{dx}\right|_{(1,0)} = - \frac{3}{2} \)
Lines: 4 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. The explanation has not been reviewed yet.
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 |
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 2026-09-26 with SymPy 1.14.0.