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