Derivative of an implicit function:

If x and y are related by rule F (x, y) = 0 such that y cannot be attained entirely or exactly in the terms of x then y is an implicit function of x.

For example: 

Here we do not obtain the unique value of y for each x.

Eg. x² - y³ + 3x²y = 0

Here y cannot be obtained entirely in the terms of x. To find dy/dx in such cases begin differentiating the given equation as it is (by using rule of composite functions)

For example: for 

If y can be expressed in terms of x entirely, then y is said to be the explicit function of x. Note that every explicit function can also be written as implicit function y - f(x) = 0.

Example: Find out dy/dx

              (i)   log(xy) = x² + y²       (ii)        x + y = sin (xy)

Solution: (i)   log (xy) = x² + y² => logx + logy = x² + y²

               Differentiating with respect to x

                    (ii)  x + y = sin (xy)

                   Differentiating with respect to x, we get

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