# Differentiation of Implicit Functions

If the variables x and y are connected by a relation of the form f(xy) = 0 and it is not possible or convenient to express y as a function x in the form y = Ï†(x), then y is said to be an implicit function of x. To find dy/dx in such a case, we differentiate both sides of the given relation with respect to x, keeping in mind that the derivative of Ï†(y) w.r.t. x is .

For example,
= cos y â‹…  = 2y
It should be noted that  = cos y but  = cos y â‹… .

Similarly, we have , whereas .

A direct formula for implicit functions

Let f(xy) = 0. Take all terms to the left side and put left side equal to f(xy). Then