Solve Implicit Function or Relation

Introduction :

The function of implicit function is related to the variables. These two variables are given by an equation. This function has not been solved explicitly. A relation between the variables of function is said to be implicit function. For example   x^2 + y^2 = 100, y is an implicit function of x and x is an implicit function of y. In this article, we shall discuss about solve implicit function or relation.

Solve Implicit Function or Relation - Problems:

Solve implicit function or relation - problem 1:

Calculate the implicit function of x and implicit function of y in the given equation    -x^2 = - 5y  .

Solution:

Given equation is        -x^2 = - 5y.                      --------------(1)

Adding by  x^2 + 5y on both side, So we get

- x^2 + 5y +  x^2 = -5y.+ x^2 + 5y

+ 5y = + x^2                     --------------(2)

Now  divided by 5 on both sides,

`(5y)/(5)` = `( x^2) /(5)` .

Implicit function of x is                    y =  `( x^2) /(5)`.

Take equation (2)    ,         5y =  x^2 

Take square root on both sides,       `sqrt(5y)` = `sqrt(x^2)` .

` sqrt(5y)`   = x .

x = `sqrt(5y)` .

Answer:     y =  `( x^2) /(5)`.  is Implicit function of x .                       

x = `sqrt(5y)` . is Implicit function of y .                               

Solve implicit function or relation - problem 2:

The implicit function function is 10xy^2 - 5y^2  = 5. Evaluate  `(dy/dx)` .

Solution:

Given implicit function is    10xy^2 - 5y^2  = 5.

Now Find the derivative of  xy^2

`d/dx`(10xy^2)    = x 20y `(dy/dx)` + 10y^2 (1).

Find the derivative of 5y^2

`d/dx`(5y^2)    =  10y `(dy/dx)` .

Find the derivative of 5 (constant)

`d/dx`(5)    = 0.

So,                     10xy^2 - 5y^2  = 5.

20xy `(dy/dx)` + 10y^2 - 10y `(dy/dx)` . = 0

Subtract by 10 y^2 on both sides,

20xy `(dy/dx)` + 10y^2 - 10y `(dy/dx)` - 10 y^2 . = 0 - 10y^2

20xy `(dy/dx)` - 10y `(dy/dx)` .=  - 10y^2

Take `dy/dx` in common

` (dy/dx)` (20xy - 10y) = - 10y^2

Divided by (20xy - 10y) on both side so we get,

` (dy/dx)`` ((20xy - 10y)/(20xy-10y))` = `((- 10y^2)/(20xy-10y))`.

` (dy/dx)`  =  `((- 10y^2)/(20xy-10y))`.

Answer:  ` (dy/dx)`  =  `((- 10y^2)/(20xy-10y))`.

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Solve Implicit Function or Relation - Practice Problems:

Solve implicit function or relation - practice problem 1:

Find the implicit function of y in the given equation    x  = 2 - 3xy .

Answer:     Implicit function of y is               x  = `((2)/(1 + 3y))`         ..

Solve implicit function or relation - practice problem 2:

Find the implicit function of x in the given equation    3y  = 9 - 3xy .

Answer:     Implicit function of x is               y  = `((3)/(1 + x))`         ..