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Differentiating a·x² + b·y² = c with respect to x treats y as a function of x, giving 2ax + 2by·(dy/dx) = 0 and hence dy/dx = −ax/(by). For a = b = 1 this is the circle, where the tangent is perpendicular to the radius.
Implicit derivative
a·x² + b·y² = c ⇒ dy/dx = −ax / (by)
dy/dx = −3/4, and the normal slope 4/3 points along the radius, as it must.
Because solving for y first may be impossible or messy, as with x³ + y³ = 6xy.