What is the $d$ used in calculus?

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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?

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1 Answer

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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.

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