The derivative of a Function

The limit that produced both the tangent slope and the instantaneous velocity is important enough to deserve its own name and its own general definition, independent of any particular geometric or physical reading.

Definition 10.2 (The derivative). Let $f$ be defined on an open interval $I$ containing the point $x_0$, and suppose

$\lim_{\Delta x\to0}\frac{f(x_0+\Delta x)-f(x_0)}{\Delta x}$

exists. Then $f$ is said to be differentiable at $x_0$, and this limit — denoted $f'(x_0)$ — is called the derivative of $f$ at $x_0$:

$f'(x_0) = \lim_{\Delta x\to0}\frac{f(x_0+\Delta x)-f(x_0)}{\Delta x} = \lim_{\Delta x\to0}\frac{\Delta y}{\Delta x}.$

More generally, for every $x$ at which this limit exists, $f'(x) = \lim_{\Delta x\to0}\dfrac{f(x+\Delta x)-f(x)}{\Delta x}$ defines $f'$ as a new function of $x$ in its own right — the derivative function. It is easy to lose sight of this: $f'$ is not just "a number attached to $f$", it is itself a full function of $x$, whose value at any particular point $x$ gives the slope of the tangent to $y=f(x)$ at $(x,f(x))$, wherever that tangent exists.

The process of computing $f'$ is called differentiation. $f$ is differentiable at $x$ if $f'(x)$ exists, and differentiable on an open interval $(a,b)$ if it is differentiable at every point of $(a,b)$.

Notations. Besides $f'(x)$ ("$f$ prime of $x$" / "$f$ dash of $x$"), the same object is written $y'$, $\dfrac{dy}{dx}$, $\dfrac{d}{dx}f(x)$, $Dy$, or $D[f(x)]$ — where $\dfrac{d}{dx}$ or $D$ is called the differential operator. The Leibniz symbol $\dfrac{dy}{dx}$ is read "derivative of $y$ with respect to $x$", or informally "dee $y$ dee $x$" / "dee dee $x$ of $y$". …