Hi guys welcome to NumberX!
Today we are going to learn about different types of notation used for representing differential of a function.
If you enjoy this video, don't forget to like it and subscribe to our channel.
Post any questions or suggestions you have, in the comments below.
In differential calculus, there is no single uniform notation for differentiation.
Instead, several different notations for the derivative of a function or variable have been proposed by different mathematicians.
The most common notations for differentiation are Leibniz’s Notation,Lagrange’s Notation
Euler’s Notation and Newton’s Notation. Let us learn these notations one by one.
Let y is a function of x which is equal to x to nth power.
In Leibniz notation first derivative of the given function is denoted by dy over dx.
And second derivative is denoted by d-two-y over dx squared.
On similar fashion, nth derivative is denoted by d-n-y over dx to the nth power.
One of the most common modern notations for differentiation is Lagrange’s notation.
In this notation, first derivative is denoted as f-prime of x.
Second derivative is denoted as f-double prime of x.
Third derivative is denoted as f-triple prime of x.
Lagrange’s notation of derivative is also called prime notation, due to use of prime sign over there.
But what about, fourth derivative or higher derivative?
It is denoted by using either roman superscript or natural number enclosed in parenthesis.
So, nth derivative is denoted by f of x and “n” at right top corner of letter “f” enclosed in parenthesis as superscript of “f”.
In Euler’s notation,
The first derivative of the function f is denoted by Dxf,
here subscript x means, function is being differentiated with respect to variable “x”.
Second derivative is denoted as Dx squared of f,
Third derivative is denoted as Dx cubed of f.
Similarly, nth derivative is denoted as Dx to nth of f.
Finally,
Newton's notation for differentiation places a dot over the dependent variable.
Here, y is function of x, hence, x is independent variable and y is dependent variable.
On top; a single dot over y denotes the first derivative of the function.
And two dots over y denotes second derivative of the function.
Similar, thing happens with third derivative also.
Due to use of dots on top of dependent variable, Newton’s notation is also called dot notation of differentiation.
But, what about higher derivatives?
In Newton’s notation, fifth derivative is denoted by “a dot beneath 5” on top of “y”.
Similarly, nth derivative is denoted by “a dot beneath n” on top of “y”.
The usefulness of each notation varies with the context, and it is sometimes advantageous to use more than one notation in a given context.
For example,
Leibniz's notation allows one to specify the variable for differentiation in the denominator. 
This is especially helpful when considering partial derivatives.
It also makes the chain rule easy to remember and recognize.
Lagrange’s notations are preferred, when a function has two independent variables.
Also, it is more compact notation than Leibniz’s notation.
Euler's notation is useful for stating and solving linear differential equations,
as it simplifies presentation of the differential equation,
which can make seeing the essential elements of the problem easier.
Newton's notation is generally used when the independent variable denotes time.
If location y is a function of t,
then single dot over y denotes velocity,
and double dots over y denotes acceleration.
This notation is popular in physics and mathematical physics.
:) Subscribe to Watch More videos like this:)
