Derivative of Implicit Functions

Implicit Function
If the independent and the dependent variables are mixed up in such a way that the dependent variable cannot be expressed in terms of the independent variable, this function is called an implicit function.

For example, {x^2} + {y^2} - x + y + 3 = 0 is an implicit function, because the dependent variable y cannot be expressed in terms of the independent variable x.

Example: Find \frac{{dy}}{{dx}} if the given implicit function is {x^2} + {y^2} = {a^2}

We have the given implicit function

{x^2} + {y^2} = {a^2}

Differentiating with respect to x, we have

\frac{d}{{dx}}{x^2} + \frac{d}{{dx}}{y^2} = \frac{d}{{dx}}{a^2}

Here, {a^2} any constant and the derivative of the constant function is zero.

\begin{gathered} 2x + 2y\frac{{dy}}{{dx}} = 0 \\ \Rightarrow 2y\frac{{dy}}{{dx}} = - 2x \\ \Rightarrow \frac{{dy}}{{dx}} = - \frac{{2x}}{{2y}} \\ \Rightarrow \frac{{dy}}{{dx}} = - \frac{x}{y} \\ \end{gathered}