Given. f(x, y) = 0

Prove. If x and y are restricted to values that satisfy f(x, y) = 0,

Proof. Let

U = f(x, y)

The total differential of U is given by

If x and y are restricted to values that satisfy f(x, y) = 0, then U = f(x, y) = 0 and dU = 0. If we replace dU in 1) by 0, and solve for dy/dx we have

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