Implicit Function

Definition

There is not only functions in the world of math. There are some times that a equation cannot be written to the form of function. We call them "implicit function", or multivariate functions.

The Derivative of Implicit Functions

It is simple to find the derivative of implicit function.

The core rule is to find the derivative to ${\displaystyle x}$ of each item in the equation, when we meet ${\displaystyle y}$, the function of ${\displaystyle x}$, use the chain rule to find the derivative again.(We consider ${\displaystyle y}$ as ${\displaystyle g(x)}$here)

In difficult problems, the most important is to follow the chain rule.

For example:

${\displaystyle (\ln y{)}^{\prime }=\frac{1}{y}{y}^{\prime }}$,

that is, for each item, find the derivative itself then the derivative the function we want.

At last, simplify the equation and put ${\displaystyle y^{\prime }}$ one side and all ${\displaystyle y}$, ${\displaystyle x}$ another side so that ${\displaystyle y^{\prime }}$ is cleared.

From Implicit Functions to Total Differential

It is a little strange that why we can find the derivative of all items in the equation.