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
- 3 See also
- 4 Bibliography
- 5 References

## Examples

- 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. ^{[2]}

## 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