Differentiation of Implicit Functions

By an implicit function we mean a function of x say y, where y cannot be expressed in terms of x only. The examples of such functions are:

(i) x² -y²  + 3X = 5y

(ii)  x² + y³ -y²  + xy =3

(iii)  x²+ xy² – y = 5

In such a type of function, differential coefficients i.e. dy/dx  is to be found by differentiating the given equation term by term as per the following steps.

Steps:

1. Differentiate both the sides of the given relation with respect to x. While differentiating the y terms not containing x multiply the same coefficient by  dy/dx.

Thus  y², will be differentiated as 2y(dy/dx)  and  3y⁵ as 15y⁴. dy/dx.

2. Transfer all the terms containing dy/dx to one side, and the terms not containing dy/dx to the other side of the equation.
3. Divide both sides by the coefficient of dy/dx, and get  there by.
4. Simplify the result by using the given relation if possible.

The following examples show how the implicit functions are differentiated.