University Mathematics Handbook (2015)

VII. Differential Calculus of Multivariable Functions

Chapter 8. Implicit Function

a.  Function is called an implicit function if it is given as a solution of equation .

b.  Theorem: If function is defined in the neighborhood of point and holds:

1.  

2.   belongs to class in the neighborhood of .

3.  

Then, there exists a neighborhood of where there exists unique function which holds and has the following properties:

a)  

b)   is continuous at

c)   is partially derivative at , and

c.  The theorem applied for a two-variable function: if function holds the conditions mentioned in b.1-3, in the neighborhood of , then there exists a neighborhood of where unique function is defined, such that , and is continuous and derivable function, the derivative of which is