We have considered exclusively functions of x that were explicitly (directly) given in terms of x. However, sometimes we deal with relationships between x and y in which it may be very complicated or even impossible to find an explicit expression for y and yet we need to find dy / dx. For example, it would be difficult to solve x³ + y³ – xy = 0 for y.

An equation of the form f (x,y) = 0, such as x³ + y³ – xy = 0, is said to express y implicitly as a function of x. The word “implicitly” is used since y is not given explicitly as a function of x. It is assumed or implied that the equation of (x,y) = 0 defines y as at least one differentiable function of x. Hence to find dy/dx, we differentiate each term with respect to x treating, y as a function of x and then separating dy/dx. This method of finding dy/dx is called implicit definition.

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