The slope of the tangent is found by

implicit differentiation and substituting (a, b)

Substituting (a, b) before differentiating

Finding dx/dy and taking the reciprocal at x = 0

Using the chord joining the origin to (a, b)

Tangent at (a,b) to (x/a)³+(y/b)³=2

Short answer

Implicit differentiation gives slope –b/a at (a,b), so the tangent is x/a + y/b = 2, i.e., bx + ay = 2ab.

Find the tangent at (a, b) to the curve (x/a)³ + (y/b)³ = 2.

Curve: (x/a)³ + (y/b)³ = 2. The point (a, b) lies on it, since 1 + 1 = 2.

Slope. Differentiate implicitly:

3x²/a³ + 3y² (dy/dx)/b³ = 0, so dy/dx = −(b³ x²)/(a³ y²).

At (a, b): dy/dx = −(b³ a²)/(a³ b²) = −b/a.

Tangent: y − b = −(b/a)(x − a). Multiply by 1/b:

(y − b)/b = −(x − a)/a, so x/a + y/b = 2.

Answer: the tangent at (a, b) is x/a + y/b = 2, i.e. bx + ay = 2ab.

✓ Answer checked by Claude Opus 5.5 (high effort)