Suppose the values of the variables x, y are linked by an equation that will be written symbolically and briefly
H (x; y) = 0 → (1)
If the function y = φ (x) defined in a certain interval <a; b> and by replacing a and in (1), the equation becomes an identity with respect to x, it is called a function implicitly determined by an equation ; in this case po H (x; y). [one]
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- 1 Examples
- 2 Bypass
- 3 See also
- 4 Bibliography
- 5 References
- 2x + 3y -5 = 0.
- 9x 2+ 4y 2 – 36, → (2), which implicitly determines the following two functions:
- y = 0.5 (36-9x 2) 5
- y = -0.5 (36-9x 2) 5
.. these equations have been obtained by solving equation (2).
However, there are cases when the proposed equation cannot be solved:
- x -8y = e x-y
- x 2+ y 2 = sin (x + y)
The general solutions of many ordinary differential equations are equations that carry functions that are implicitly definable, but not explicit. 
Let be the equation x 3 + y 3 = 3axy, we are going to derive considering y = y (x) and using the derivative rule of the compound function.
3x 2 + 3y 2 y ‘= 3ay + 3axy’ and finally
y ‘= (ay-x 2 ) ÷ (y 2 – ax)
Given the equation sin x + cos y = a, find y ‘, we have
cosx + seny · y ‘= 0
y ‘= cosx ÷ sin y