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I know the letter $d$ is commonly used in calculus and represents a derivative. Does this $d$ act as a variable that can be simplified or as a function of another variable?

dmtri
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RK01
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    $d/dx$ is an operator. The domain of the operator is set of functions with derivatives at every point of the domain, with the range being another set of functions. –  Jul 01 '15 at 13:00
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    Berkeley called these or somethings similar "ghosts of departed quantities" – Henry Jul 01 '15 at 13:04

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While the $d$ in Leibnitz's notation is often used as a variable in operations on derivatives (see below), this is merely a convenience of the notation. The $d$ itself simply stands to indicate which is the independent variable of the derivative ($x$) and which is the function for which the derivative is taken ($y$). $\frac{d}{dx}$ itself is an operator on function $y$.

The second derivative (the derivative of a derivative), is written as if it is the product of two derivatives $\frac{d}{dx}\times\frac{dy}{dx}=\frac{d^2y}{dx^2}$ (with parentheses being assumed in the denominator $\frac{d^2y}{(dx)^2}$)

The chain rule can also be understood as a form of multiplication, $\frac{dy}{dx}=\frac{dy}{du}\times\frac{du}{dx}$, with the $u$ canceling.

However, it must always be held in consideration that the reason for these equalities is not derived from such symbol manipulation but rather from rigorous proof. They are convenient notations which are understood to represent a deeper meaning.

Isaac Scheinfeld
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    You might also be interested in this answer that I gave to a similar question, mainly related to the last part of this answer regarding the lack of rigour in the appealing symbol manipulation. – String Jul 01 '15 at 13:23
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    Thanks! Your answer in regard to Bos's work on contradiction in Leibniz's notation is also fascinating. – Isaac Scheinfeld Aug 23 '15 at 12:11