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.
- 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.
- Transfer all the terms containing dy/dx to one side, and the terms not containing dy/dx to the other side of the equation.
- Divide both sides by the coefficient of dy/dx, and get there by.
- Simplify the result by using the given relation if possible.
The following examples show how the implicit functions are differentiated.