Subject: Calculus

Leibniz's Notation for Differentiation

During the 17th-century, a German philosopher and mathematician in the name of Gottfried Wilhelm Leibniz discovered Calculus. While Newton's discovery on calculus is based on limits and concrete reality, Leibniz focused more on the infinite and the abstract.[1]
Their dissimilarities in paths taken in discovering calculus also reflect the type of notations they used in describing derivatives. Newton preferred dots placed at the top of the function being differentiated while Leibniz is happy to denote it with something like "dx" or "dy" for functions or variables x and y. He uses such symbols to represent "infinitely small" increments of x and y, just as \delta x and \delta y represent finite increments of x and y.[2] As an illustration, consider the function y=f(x). We already knew that in Newton's notation, this is denoted as \dot{y}. Then according to Leibniz's notation, the first order derivative of f with respect to x is written as

\dot{y}=\frac{dy}{dx}

Higher Order Derivatives

In Newton's notation, we can write the second derivative of the function y=f(x) as \ddot{y}. In Leibniz's notation, this is written as

\ddot{y}=\frac{d^{2}y}{dt^{2}}.

For its third order derivative, Leibniz symbolizes it as

f'''(x)=\frac{d^{3}y}{dx^{3}}

and for the nth-order derivative, this is given as

f^{n}(x)=\frac{d^{n}y}{dx^{n}}

Example #1

Find the corresponding Leibniz's notation of the following derivatives written in Newton's notation (f(z)=x).

\ddot{x}

\frac{d^{2}x}{dz^{2}}

Example #2

Find the corresponding Leibniz's notation of the following derivatives written in Newton's notation (f(r)=z).

\dot{z}

\frac{dz}{dr}

Example #3

What is the Leibniz notation for a function in third order derivative?

\frac{d^3}{dx^3}f(x)

Example #4

What is the Leibniz notation for a function in third order derivative?

\frac{d^4}{dx^4}f(x)

Example #5

What is the Leibniz notation for a function in third order derivative?

\frac{d^5}{dx^5}f(x)

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