Leibniz's notation

In calculus, Leibniz's notation, named in honor of the 17th-century German philosopher and mathematician Gottfried Wilhelm Leibniz, is a notation for differentiating which uses the symbols dx and dy to represent infinitely small (or infinitesimal) increments of x and y, respectively, just as Δx and Δy represent finite increments of x and y, respectively.

Given:

$y=f(x).$

Then the derivative in Leibniz's notation for differentiation, can be written as

${\frac {dy}{dx}}\,{\text{ or }}{\frac {d}{dx}}y\,{\text{ or }}{\frac {d{\bigl (}f(x){\bigr )}}{dx}}.$

Representing dy with respect to dx, or with respect to any other variable on the other side of the function (${\frac {dy}{da}}$). If $x = f(x)$:

${\frac {dx}{dy}}\,{\text{ or }}{\frac {d}{dy}}x\,{\text{ or }}{\frac {d{\bigl (}f(y){\bigr )}}{dy}}.$

Representing dx with respect to dy.

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A common alternative is Lagrange's notation

${\frac {dy}{dx}}\,=y'=f'(x).$

Another alternative is Newton's notation, often used for derivatives with respect to time (like velocity), which requires placing a dot over the dependent variable (in this case, x):

${\frac {dx}{dt}}={\dot {x}}.$