Leibniz notation

Named in honor of Gottfried Leibniz, a seventeenth century philosopher and mathematician, Leibniz notation is one of the two most popular ways to represent the derivative of a function. (The other representation being Lagrange notation.)

Suppose we have a variable y that represents f, a function of the independent variable x:

y = f(x)

The derivative of the function f can be written using Leibniz notation as:

    \[\frac{dy}{dx} \quad \text{or} \quad \frac{d}{dx}y\quad \text{or} \quad \frac{d(f(x))}{dx}\]

The first two forms can be rewritten in Lagrange notation like so:

    \[\frac{dy}{dx} \equiv \frac{d}{dx}y \equiv y'\]

And the third form in Lagrange notation is:

    \[\frac{d(f(x))}{dx} \equiv f'(x)\]

For higher derivatives, the Leibniz notation for the nth derivative is:

    \[\frac{d^ny}{dx^n} \quad \text{or} \quad \frac{d^n(f(x))}{dx^n}\]

For example the second derivate of y or f(x) would be:

    \[\frac{d^2y}{dx^2} \quad \text{or} \quad \frac{d^2(f(x))}{dx^2}\]




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