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.
\[\begin{gathered} 2x + 2y\frac{{dy}}{{dx}} = 0 \\ \Rightarrow 2y\frac{{dy}}{{dx}} = – 2x \\ \Rightarrow \frac{{dy}}{{dx}} = – \frac{{2x}}{{2y}} \\ \Rightarrow \frac{{dy}}{{dx}} = – \frac{x}{y} \\ \end{gathered} \]