# Implicit Differentiation Formula

## Implicit Differentiation Formula

The method of calculating the derivative of an implicit function is known as implicit differentiation. In other words, this method is used to determine the implicit derivative. Functions come in two forms: explicit functions and implicit functions. The dependent variable “y” is on one of the sides of the equation in an explicit function, which has the formula y = f(x). However, ‘y’ does not always have to be on one side of the equation. Even when “y” is not one of the sides, we can still solve it to form the explicit function. This type of function is known as an implicit function. The process of differentiating an implicit equation with respect to the desired variable x while treating the remaining variables as undetermined functions of x is known as Implicit Differentiation Formula. Any of the two well known approaches is used to differentiate an implicit function. In the first approach, the implicit equation is differentiated for y and the implicit equation is explicitly written in terms of x. Only when y is simply expressed in terms of x is this strategy considered to be useful. The second approach differentiates both components of the implicit equation with respect to x and views y as a function of x. To determine the value of dy/dx, the resulting equation is solved. An implicit equation in Mathematics is a relation of the type R(x1,… ,xn)=0, where R is a function of several variables (often a polynomial). The implicit equation of the unit circle, for instance, is x2+ y2 – 1 =0. An implicit function is one that is defined by an implicit equation that connects one of the variables, referred to as the function’s value, to the others, referred to as the arguments. The implicit function theorem establishes the requirements for specific types of implicit equations to create implicit functions, namely those that result from equating to zero continuously differentiable multivariable functions.

## What is Implicit Differentiation?

The chain rule is used to differentiate implicitly defined functions in calculus using a technique known as Implicit Differentiation Formula. It is typically not possible to solve an implicit function y(x) directly for y and then differentiate it, since it is specified by an equation R(x, y) = 0. Instead, one can directly obtain the derivative in terms of x and y by fully differentiating R(x, y) = 0 with respect to x and y and then solving the ensuing linear equation for dy / dx. Even though the original equation can be explicitly solved, the formula that results from total differentiation is typically far more easy and user-friendly. Differentiating an implicit function is known as Implicit Differentiation Formula. f(x, y) = 0 is a form that can be used to express an implicit function. Specifically, it is difficult to solve for “y” or convert it to the form y = f (x). One can consider the problem of determining dy/dx given the function xy = 5. There are two ways to determine dy/dx: Without solving for y and after solving for y. The implicit function is transformed into the explicit function in method 1 and the power rule is used to find the derivative. However, in approach 2, Implicit Differentiation Formula is applied by treating y as a function of x, which allows differentiating both sides with respect to x. However, it is impossible to write some functions explicitly (Method 1), such as xy + sin (xy) = 0. Only Implicit Differentiation Formula (Method 2) can be used to find the derivative in these circumstances. All of the variables in the equation, which may include many independent and dependent variables, must be taken into account in order to differentiate the implicit function. One can use partial differentiation to separate the expression from other variables.

## Implicit Derivative

The implicit derivative is the derivative obtained through the method of Implicit Differentiation Formula. For instance, the Implicit Differentiation Formula is the derivative found in Method 2 (in the example above) that was initially equal to dy/dx = -y/x. This is due to the direct differentiation of the implicit function xy = 5 rather than solving for y. Typically, an implicit derivative has terms for both x and y. When both variables in an equation are non-linear, one cannot get an explicit form in which they may express one variable in terms of the other, and thus get an equation having an implicit form.  For instance, it is impossible to express a circle’s equation explicitly as a single equation because it is stated as an equation in which both variables appear squared. Given that the two variables are not linear, an Implicit Differentiation Formula means the act of differentiating an implicit form. Their characteristics can help students better comprehend implicit functions. For the implicit f(x), y = f(x) cannot be written. The interpretation of an implicit function is always f(x, y) = 0. The implicit function consists of various variables. The implicit function is written using the dependent and independent variables. The vertical line formed along the graph of an implicit function passes through a number of points.

### Implicit Differentiation and Chain Rule

When determining the implicit function’s derivative, the chain rule of differentiation is crucial. According to the chain rule, (f(g(x)) = (f’ (g(x)) g’ (x). The chain rule enters the picture whenever one encounters the derivative of y terms with respect to x, and as a result of the chain rule, students can multiply the actual derivative (by derivative formulae) by dy/dx. In other words, it is essential to include dy/dx whenever y is being differentiated. It is advised to review some examples repeatedly because they are crucial for performing implicit differentiation. In issues involving linked rates or curves with implicit forms that are given in rectangular form, an Implicit Differentiation Formula is frequently applied to determine the slope of the tangent line to the curve. In such a related rate problem, students are given one variable’s rate of change and must determine the rate at which another variable is changing. To achieve this, one must take the two actions listed below:

Identify the relationship between the two variables, which is typically an implicit one.

Find a link between the two rates using the Implicit Differentiation Formula to find the unknown rate.

### How to Do Implicit Differentiation?

Since an implicit function does not have the form y = f(x), but rather has the form f(x, y) = 0, one cannot begin the process of implicit differentiation directly with dy/dx. It is important to remember that before mastering the method of implicit differentiation, students should be familiar with the derivative rules such as the power rule, product rule, quotient rule, chain rule, etc. For finding the Implicit Differentiation Formula dy/dx for a function, these stages are all discussed.

Step 1: Distinguish between each term on both sides in relation to x.

Step 2: Use the chain rule and the derivative formulas to find the derivatives.

(Using the derivative formulas, the x terms should be directly differentiated; however, when differentiating the y terms, students should multiply the actual derivative by dy/dx.)

Step 3: Solve the equation for dy/dx using the common factor of dy/dx. This gives the Implicit Differentiation Formula.

### Implicit Differentiation Formula

The procedures for performing implicit differentiation have been shown in the study materials by Extramrks. Students should follow the processes outlined to determine the Implicit Differentiation Formula. Differential equation solutions frequently take the form of an implicit function. The Implicit Differentiation Formula is also used in Economics when the level set R(x, y) = 0 represents an indifference curve for the amounts x and y consumed of two items. The absolute value of the Implicit Differentiation Formula dy / dx is read as the marginal rate of substitution of the two goods: how much more y one needs to receive to be indifferent to a loss of one unit of x. In economic theory, it is common for some function, such as a utility function or a profit function, to be maximised with respect to a choice vector x, even though the objective function is not restricted to any precise functional form. The implicit function theorem ensures that each component of the optimal vector x* of the choice vector x has an implicit function defined by the optimization’s first-order requirements. The supply functions of various items and the labour demand function are often the implicit functions that result when profit is being maximised. The labour supply function and the demand functions for various items are often the implicit functions that result when utility is maximised.

### Implicit Differentiation of Inverse Trigonometric Functions

Finding the derivatives of inverse trig functions is made easier by the procedure of implicit differentiation. Students can use the Implicit Differentiation Formula to determine the derivative of y = tan-1 x. The derivatives of any inverse function can be determined using the method of implicit differentiation. The Implicit Differentiation Formula must also be done using all derivative formulas and methods. The process of implicit differentiation does not have a set formula. Instead of ignoring the chain rule and calculating the resulting equation for dy/dx, one can simply differentiate the function on both sides. One can simply multiply the derivative by dy/dx whenever they want to differentiate something with y. Inverse functions are a typical kind of implicit function. Some functions do not have a specific inverse function. If g is a function of x that has an exclusive inverse, then g-1, the inverse function of g, is the only function that can provide a solution to the equation.

### Implicit Differentiation Examples

Students can find various real life examples that make use of the Implicit Differentiation Formula. Understanding implicit functions is essential in order to grasp calculus’s complex ideas. In both architecture and material science, implicit functions have a wide range of applications. Implicit functions are the fundamental building blocks of calculus. Students shall be able to comprehend the implicit functions to be good at Calculus. The Implicit Differentiation Formula is used to express rates of change. Numerous applications of calculus can be found, such as creating a differential equation with the unknown function y=f(x) and its derivative. These equations’ solutions occasionally show how and why particular variables change.

### Practice Questions on Implicit Differentiation

In order to comprehend the Implicit Differentiation Formula clearly, it is necessary to practice answering a variety of questions. For practice, Extramarks offers a number of practice questions on the Implicit Differentiation Formula.