Mr. Integral
Math notes/Calculus III/Partial Derivatives and Local Change

Partial Derivatives and Differentiability — Cheat sheet

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Core rule

L(x,y)=f(a,b)+fx(a,b)(x−a)+fy(a,b)(y−b).L(x,y)=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b).

Watch for

Partials alone do not imply continuity or differentiability. Continuous first partials nearby are a sufficient condition for a valid linearization.