# Function implicitly determined by an equation

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]

## Summary

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• 1 Examples
• 2 Bypass
• 4 Bibliography
• 5 References

## Examples

• 2x + 3y -5 = 0.
• 9x 2+ 4y 2 – 36, → (2), which implicitly determines the following two functions:
1. y = 0.5 (36-9x 25
2. y = -0.5 (36-9x 25

.. these equations have been obtained by solving equation (2).

However, there are cases when the proposed equation cannot be solved:

1. x -8y = e x-y
2. 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. 

## Derivation

Example 1

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)

Example 2

Given the equation sin x + cos y = a, find y ‘, we have

cosx + seny · y ‘= 0

y ‘= cosx ÷ sin y