Implicit differentiation is a tool used to analyze functions that cannot be conveniently put into a form $y=f(\mathbf{x})$ where $\mathbf{x} = (x_1, x_2, ..., x_n)$ To use implicit differentiation meaningfully, you must be certain that your function is of the form $f(\mathbf{x})=0$ (it can be written as a level set) and that it satisfies the implicit function theorem ($f$ must be continuous, its first partial derivatives must be continuous, and the derivative with respect to the implicit function must be non-zero). To actually differentiate implicitly, we use the chain rule to differentiate the entire equation.

Example: The first step is to identify the implicit function. For simplicity in the example, we will assume $f(x,y)=0$ and $y$ is an implicit function of $x$ Let $f(x,y)=x^2 + y^2 + xy =0$ (Since this is a two dimensional equation, all one has to check is that the graph of $y$ may be an implicit function of $x$ in local neighborhoods.) Then, to differentiate implicitly, we differentiate both sides of the equation with respect to $x$ We will get $$2x + 2y\cdot \frac{dy}{dx} + x\cdot 1\cdot\frac{dy}{dx} + y = 0$$ Do you see how we used the chain rule in the above equation ? Next, we simply solve for our implicit derivative $\frac{dy}{dx}=-\frac{2x+y}{2y+x}$ Note that the derivative depends on both the variable and the implicit function $y$ Most of your derivatives will be functions of one or all the variables, including the implicit function itself.

[better example and ?multidimensional? coming]