Differentiation of Implicit Function

A function developed for differentiation of functions comprising variables but not easily written in y = f(x)  is called an implicit function (x). If x2 + y2 + 4xy + 25 = 0, then the dependent and independent variables “y” cannot be easily separated to describe it as a function of “y = f(x),” then it is an example of an implicit function (x). 

To better grasp implicit function, we need to understand it better. So let’s look at some real-world examples to better understand the concepts of implicit function and implicit function differentiation.

Implicit Function Theorem

The implicit function theorem is a method used in mathematics to turn relations into functions of many real variables. It is most commonly used in multivariable calculus. It accomplishes this by displaying the relationship as a function graph. Although a single function whose graph can represent the entire connection may not exist, such a function may exist on a subset of the relation’s domain. The implicit function theorem establishes a necessary condition for the existence of a function.

The theorem states that, given a system of m equations fi (x1,…, xn, y1,…, ym) = 0, I = 1,…, m (often abbreviated into F(x, y) = 0), the m variables yi are differentiable functions of the xj in some neighbourhood of the point under a mild condition on the partial derivatives (concerning the yis). Because these functions are rarely given in the closed form, the equations implicitly define them, hence the theorem’s name .

Differentiation of Implicit Function

Differentiation has been used to define the term “implicit function,” which refers to functions with many variables that are difficult to differentiate. The chain rule of differentiation of functions is used to determine implicit functions. 

To better comprehend implicit functions, let’s first look at explicit functions. For example, y = f(x) is a common formula for manipulating and expressing basic linear equations in x and y, and it is referred to as an explicit function. In this case, it’s easy to tell the difference between the dependent variable y and the independent variable x. 

To differentiate the implicit function, one must consider all of the variables in the equation, which can include more than one independent and dependent variable. We can isolate the expression from other variables by utilising partial differentiation. 

Two easy steps are all that is required to differentiate an implicit function. When f(x, y) = 0 is decomposed into its components, the first step is to determine which two independent variables are zero. Then, algebraically shift the variables to determine the expression’s dydx  value. 

Steps to Compute the Derivative of an Implicit Function

• If y is a dependent variable and x is an independent variable in a given implicit function, then (or the other way around). 
• Calculate the derivative of each term in the equation, taking into account the independent variable (it could be x or y). 
• The chain rule of differentiation must be used once we have differentiated. 
• If higher-order derivatives are required, solve the resulting equation for dydx  (or dxdy  in the same way).

Properties of Implicit Function

It is helpful to know the following properties to understand implicit functions better. 

• It is impossible to write y = f(x) for the implicit function f(x). 
• When the function is implicit, it is always interpreted as f(x,y) = 0. 
• There are numerous variables involved in the implicit function. 
• The dependent and independent variables are used to formulate the implicit function. 
• Several points are crossed by the vertical line drawn along the graph of an implicit function.

Conclusion

An implicit function is a vital topic to be studied to understand the deep concepts of calculus. Implicit functions have a vast scope and use in architecture and material science. To understand calculus, implicit functions are the building blocks. Through this topic, we will be able to understand the implicit functions.

Mathematicians use derivatives to express rates of change in calculus. Calculus is used extensively in various ways, including formulating a differential equation that includes an unknown function y=f(x) and its derivative. Sometimes, the solutions to these equations reveal how and why specific variables change.