Subject: Calculus

# Differential Operator

In calculus, we define differential operator as an operator that indicates the operation of differentiation or simply the process of calculating a derivative. Suppose we have a variable y = f(x) which indicates that the independent variable x is and the dependent variable is y. We can therefore write the derivative of y in different notations which I will be discussing in the next part.

## Notations

Generally, the notations for the derivatives of y = f(x) can be written as follows,

Please do take note that the notation dy/dx is not a ratio. This was introduced by Leibniz which is simply a synonym for f’(x). Nonetheless, it is a very useful and suggestive notation, especially when used in conjunction with increment notation.

We can also rewrite the definition of a derivative in terms of Leibniz notation which takes the form of,

If we want to indicate a value of a derivative dy/dx in Leibniz notation at a specific number a, we therefore write it as

which is just a synonym of f’(a).

In higher calculus, you will frequently see a differential operator called the Laplacian operator. This is defined by

The symbols Δ and ∇^2denotes the divergence of the gradient of a function on Euclidean space.

## Properties of Differential Operator

Differential operators are linear. When we say linear, a differential operator follows the sum rule of differentiation and the constant factor rule in differentiation.

To expound that idea, suppose we have functions f and g and constants a and b. Now, if we have this expression

by sum rule of differentiation, we can express the above equation as

Furthermore, by constant factor rule in differentiation, we could further simplify the above equation as

So we now have,

In multivariable functions, differential operators may be used to only one variable of the function. This is the basic foundation of partial differential equations. In example, if we have z = f(x, y), we denote the partial differential operators as

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