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Implicit Function
Implicit function is a function defined for differentiation of functions containing the variables, which cannot be easily expressed in the form of y = f(x). The function of the form g(x, y)=0 or an equation, x^{2} + y^{2}+ 4xy + 25 = 0 is an example of implicit function, where the dependent variable 'y' and the independent variable 'x' cannot be easily segregated to represent it as a function of the form y = f(x).
Further, we also need to also learn about the explicit function, for a better understanding of implicit function. Let us learn more about implicit function and the differentiation of implicit function, with the help of examples.
1.  What Is An Implicit Function? 
2.  Derivative of Implicit Function 
3.  Properties Of Implicit Function 
4.  Examples on Implicit Function 
5.  Practice Questions 
6.  FAQs on Implicit Function 
What Is An Implicit Function?
Implicit function is a function with multiple variables, and one of the variables is a function of the other set of variables. A function f(x, y) = 0 such that it is a function of x, y, expressed as an equation with the variables on one side, and equalized to zero. An example of implicit function is an equation y^{2} + xy = 0. Also, a function f(x, y, z) = 0 such that one variable is dependent on the other two variables, is an implicit function.
The function which can be easily written as y = f(x) with the y variable on one side and the function of x on the other side, is called an explicit function. But in an implicit function, the x and the y variable cannot be written in the form y = f(x), and an implicit function has more than one solution for the given function. An expression of an implicit function contains two or more than two variables, written and expressed in the form of an equation f(x, y) = 0 or g(x, y, z) = 0, with the expression on the lefthand side of the equation containing the variables, constants, coefficients, and equalized to zero.
The relation y = f(x) is an explicit function, with y represented in terms of x. Further in a implicit function, the relationship between x and y is expressed as g(x, y) = 0, where y is an implicit function of x. More specifically the value of x defines the value of y such that on substitution of x, y in the expression on the lefthand side, equalizes it to zero. The expressions y = x^{2}, y = ax + b, y = \(\sqrt x\), are all examples of explicit functions, and the expressions ax^{2} + bxy  y = 0, x^{2}  y^{2} = 0, e^{y} + x  y + log y = 0, are the examples of implicit functions.
Derivative of Implicit Function
The concept of implicit function has been defined with reference to differentiation, where the differentiation of functions involving many variables is difficult. The differentiation of implicit functions involves chain rule of differentiation of functions.
Before looking at implicit function let us understand more about explicit function. Generally simple linear equations in x and y can be manipulated and expressed as y = f(x) and is called an explicit function. Here the dependent variable y and the independent variable x can be clearly distinguishable and is differentiated with respect to x.
The implicit function is of the form f(x, y, z) = 0 and can have more than one variables, which cannot be separated as a dependent variable and independent variable for differentiation. Here we use partial differentiation to differentiate the entire expression with respect to one particular variable.
The differentiation of implicit function involves two simple steps. First differentiate the entire expression f(x, y) = 0, with reference to one independent variable x. As a second step, find the dy/dx of the expression by algebraically moving the variables. The final answer of the differentiation of implicit function would have both variables. Let the understand this with the help of an example.
Example
x^{2}+ xy + y = 0
Step  1: Let us differentiate this expression with respect to x, with both the variables on the left side.
2x + x.dy/dx +y.1 + dy/dx = 0
Step  2: Algebraically segregate the variables to find dy/dx of the expression.
x.dy/dx + dy/dx = (2x + y)
dy/dx(x + 1) = (2x + y)
dy/dx = (2x + y)/(x + 1)
Properties of Implicit Function
The following are some of the important properties of implicit function, which are helpful in a better understanding of this function.
 The implicit function cannot be expressed in the form of y = f(x).
 The implicit function is always represented as a combination of variables as f(x, y) = 0.
 The implicit function is a nonlinear function with many variables.
 The implicit function is written both in terms of the dependent variable and independent variable.
 The verticle line draw through the graph of an implicit function cuts it across more than one point.
Related Topics
The following topics are helpful to understand the concept of implicit function.
Examples on Implicit Function

Example 1: Find the derivative of the implicit function x^{2} + y^{2} + 4xy + 7 = 0, and find dy/dx.
Solution
The given function is x^{2} + y^{2} + 4xy + 7 = 0
Here we differentiate the function with respect to x.
d/dx.x^{2 }+ d/dx.y^{2} + d/dx.4xy + d/x.7 = 0
2x + 2y.dy/dx + 4(x.dy/dx + y.d/dx.x) + 0 = 0
2x + 2y.dy/dx + 4(x.dy/dx + y) = 0
2x + 2y.dy/dx + 4x.dy/dx + 4y = 0
2x + 4y + (4x + 2y).dy/dx = 0
2(x + 2y) + 2(2x + y).dy/dx = 0
(x + 2y) + (2x + y).dy/dx = 0
(2x + y).dy/dx = (x + 2y)
dy/dx = (x + 2y)/(2x + y)
Therefore, the derivative of the implicit function is (x + 2y)/(2x + y).

Example 2: Find the derivative of the implicit function x + Sinxy  y = 0
Solution:
The given function is x + Sinxy  y = 0
Here we differentiate the function with respect to x.
d/dx.x + d/dx.Sinxy  d/dx.y = 0
1 + Cosxy.(d/dx.xy)  dy/dx = 0
1 + Cosxy.(x.dy/dx + y.d/dx.x)  dy/dx = 0
1 + Cosxy(x.dy/dx + y)  dy/dx = 0
1 + xCosxy.dy/dx + yCosxy  dy/dx = 0
1 + yCosxy + dy/dx(xCosxy  1) = 0
dy/dx(xCosxy  1) = (1 + yCosxy)
dy/dx = (1 + yCosxy)/(xCosxy  1)
dy/dx = (1 + yCosxy)/(1  xCosxy)
Therefore, the derivative of the given implicit function is (1 + yCosxy)/(1  xCosxy).
FAQs on Implicit Function
What Is Implicit Function?
Implicit function is a function of form f(x, y) =0, which has been defined to easily facilitate the differentiation of an algebraic function. The implicit function has the variables, coefficients, constants as an equation on the lefthand side, which has been equalized to zero.
What Are the Examples of Implicit Functions?
The expressions ax^{2} + bxy  y = 0, x^{2}  y^{2} = 0, e^{y} + x  y + log y = 0, are the examples of implicit functions.
How Do We Know If A Function Is an Implicit Function?
A function that cannot be easily represented in the form y = f(x) is an implicit function. Also, an implicit function cannot be differentiated easily to find the dy/dx on the lefthand side and the derivative of it on the righthand side, in simple steps.The implicit function is generally of the form f(x, y)=0, or g(x, y, z)=0, and it contains all the variables, coefficients, constant on the lefthand side of the equation, and is equalized to zero.
What Is the Difference Between Explicit Function And Implicit Function?
The function which can be expressed in the form of y = f(x) is an explicit function, and the function of form f(x, y) = 0, is an implicit function. The implicit function has been defined to facilitate the differentiation of functions. The functions which can be easily differentiated to find the dy/dx on the left hand side and its derivative on the right hand side is an explicit function, and the function containing the dy/dx, along with the variables, coefficients can be termed as ain implicit function.
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