ImplicitDifferentiation 2002-02-25 01:20:47 2003-10-30 16:36:20 Definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here 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. \textbf{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]