Implicit Differentiation Formula

About Implicit Differentiation Formula

Implicit differentiation is the process of getting the derivative of an implicit function. The two sorts of functions are explicit and implicit functions. In an explicit function of the form y = f, the dependent variable "y" is on one of the sides of the equation (x). It is not, however, necessary to have 'y' on one side of the equation. Take the following functions, for example: x²+ y = 2 and xy + sin (xy) = 0

Even though 'y' is not one of the equation's sides, we can still solve it to write it as y = 2 - x², and it is an explicit function. However, we cannot simply calculate the equation for 'y' in the second situation, and this form of function is known as an implicit function.

What is Implicit Differentiation and How Does It Work?

The process of differentiating an implicit function is known as implicit differentiation. A function that can be stated as f(x, y) = 0 is known as an implicit function. It cannot be simply solved for 'y' (or) converted into the form y = f. (x). Consider the problem of determining dy/dx given the function xy = 5. Let's look for dy/dx using two different approaches: (i) Solving for y (ii) Not solving for y

Method 1: xy = 5
y = 5/x
y = 5x^-1
Differentiating both sides in relation to x:
dy/dx = 5(-1x^-2) = -5/x²
Method 2:
xy = 5
Differntiating both sides in relation to x:
d/dx (xy) = d/dx(5)
Using product rule on left hand side, we have
x d/dx(y) + y d/dx(x) = d/dx(5)
x (dy/dx) + y × 1 = 0
x(dy/dx) = -y
dy/dx = -y/x
From xy = 5, we can write y = 5/x.
dy/dx = -(5/x)/x = -5/x²

In Method 1, we used the power rule to convert the implicit function to an explicit function and find the derivative. However, in approach 2, we differentiated both sides with respect to x by considering y as a function of x, which is known as implicit differentiation. However, some functions, such as xy + sin (xy) = 0, cannot be written as an explicit function (Method - 1). In such instances, the only option to get the derivative is to use implicit differentiation (Method - 2)