Implicit Functions The equation y = mx + b is an explicit function –y is considered the dependent variable –x is the independent variable The equation f(x,y,m,b) = 0 is an implicit function –The relationships between the variables and the parameters are implicitly present in the equation but not explicitly stated

Derivatives from Implicit Functions In many circumstances, it will be helpful to compute derivatives directly from implicit functions If f(x,y)=0, then its total differential of f 1 dx + f 2 dy = 0 Thus,

Production Possibility Frontier Earlier example: 2x 2 + y 2 = 225 Can be rewritten: f(x,y) = 2x 2 + y = 0 Then, the opportunity cost trade-off between x and y is Because f x = 4x and f y = 2y

Implicit Function Theorem It may not always be possible to solve implicit functions of the form g(x,y)=0 for unique explicit functions of the form y = f(x) –Mathematicians have derived the necessary conditions –In many economic applications, these conditions are the same as the second-order conditions for a maximum (or minimum)