Definition of Differentiation:
We already discussed what derivatives are and how to solve derivatives and finally, we converted it in terms of limits.
The derivative definition of a function f(x) with respect to independent variable x is given as:
$$ \frac{Δy}{Δx} = \lim\limits_{ Δx \to 0} \frac{f(x+Δx) − f(x)}{Δx} = f'(x) $$
We write it as f'(x) which is the symbol of derivative notation.
What is derivative notation?
Derivatives are calculated by using the differentiation method on functions and that derivatives are denoted in a mathematical form which is known as notation for differentiation or derivative notation.
There is not a single notation for writing derivatives. In fact, different mathematicians introduced their own notation for expressing the derivative of a function in mathematical terms.
Let’s see the most common and most probably used derivative notations.
Leibniz Notation:
The first-ever notation was introduced by Gottfried Leibniz which is the most commonly used notation.
In Leibniz notation, the derivative of a function is expressed as d/d(x).
Consider a function:
$$y = f(x)$$
The symbol of derivative is denoted as :
$$ \frac{dy}{dx} = \frac{d}{dx}f(x) = \frac{df}{dx}(x) = \frac{df(x)}{dx}$$
The second derivative of "y" w.r.t "x" is :
$$ \frac{d (\frac{dy}{dx})}{dx} = \frac{\frac{ddy}{dx}}{dx} = \frac{ddy}{dx^2} = \frac{d^2y}{dx^2} $$
Hence,
The derivative notation of second derivative or above are denoted as:
$$ \frac{d^2y}{dx^2},\frac{d^3y}{dx^3},\frac{d^4y}{dx^4},... $$
Or in general, nth derivative is:
$$ \frac{d^ny}{dx^n} $$
Leibniz notation is most commonly used in integrals, multivariable and partial derivatives but in single variable derivatives, there is more comfortable notation introduced which is given below.
Lagrange Notation:
The derivative in Lagrange notation is detonated by a prime (') sign on a function. Consider we have a function:
$$ y = f(x) $$
The derivative is denoted as :
$$y' = f'(x)$$
It is mostly used in single variable derivatives for comfortable notation. In the second derivative, it is denoted by the double prime sign.('')
$$ y'' = f''(x) $$
Similarly,the derivative notation for 3rd,4th or more
$$ f'(x), f''(x), f'''(x) f''''(x),...$$
In general, lagrange notation of writing derivatoves will
$$ y^n = f^n (x)$$
Where n = number of primes(').
Newton's notation:
Newton’s notation is also called as dot-notation for differentiation. This notation is denoted by() on a function.
We have a function
$$ y = f(x) $$
The derivative of a function is written as:
$$ \dot y = \dot f(x) $$
Similarly, the second derivative is written as:
$$ \ddot y = \ddot f(x) $$
Newton notation of writing derivatives is mostly used in physics or science where derivatives are used in the real world.