# What is the ‘implicit function theorem’?

Please give me an intuitive explanation of ‘implicit function theorem’. I read some bits and pieces of information from some textbook, but they look too confusing, especially I do not understand why they use Jacobian matrix to illustrate this theorem.

Let’s use a simple example with only two variables. Assume there is some relation $f(x,y)=0$ between these variables (which is a general curve in 2D). An example would be $f(x,y) =x^2+y^2-1$ which is the unit circle in $\mathbb{R}^2$. Now you are interested to figure out the slope of the tangent to this curve at some point $x_0,y_0$ on the curve [with $f(x_0,y_0)=0$].
What you can do is to change $x$ a little bit $x = x_0 + \Delta x$. You are interested then how $y$ changes ($y= y_0 + \Delta y$); remember that we are interested in points on the curve with $f(x,y)=0$. Using Taylor expansion on $f(x,y)=0$ yields (up to lowest order in $\Delta x$ and $\Delta y$)
The slope is thereby given by As $\Delta x \to 0$ higher order terms in the Taylor expansion (which we neglected) vanish and $\frac{\Delta y}{\Delta x}$ becomes the slope of the curve implicitly defined via $f(x,y)=0$ at $(x_0,y_0)$.