]> 6.1 Differentiability, the Tangent Line-Linear Approximation

6.1 Differentiability, the Tangent Line-Linear Approximation

A function $f$ of one real variable is said to be differentiable at argument $x$ , if its graph looks like a straight line for arguments in any open interval including $x$ . (An open interval is one that does not contain its endpoints.)

Its derivative at $x$ is the slope of that line.

(To be more precise, for whatever positive criterion of nearness you choose however small, there is an open interval containing $x$ so that for every $x '$ in the interval other than $x$ itself, the difference between $f ( x ' ) − f ( x ) x ' − x$ and the slope of that line is less than that criterion.) Note

The line that $f$ resembles near argument $x$ is called the tangent line to $f$ at argument $x$ and the linear function it represents is called the linear approximation to $f$ at argument $x$ .

The slope of the tangent line at $x$ is given by $d f d x$ on that line for any two point $P 2$ and $P 1$ on it, with $P 1 = ( P 1 x , P 1 y )$ and $P 2 = ( P 2 x , P 2 y )$

$d f = P 2 y − P 1 y$
$d x = P 2 x − P 1 x$

We use the notation $d x$ and $d f$ to denote changes in the corresponding variables that are so small that we can assume the linear approximation to $f$ (and to any other function involved in the definition of $f$ ) is exacty satisfied (and if there is no such distance create one in your imagination).

Changes of this sort are called differentials . The derivative of $f$ at argument $x$ is usually written as $d f d x$ or $d f ( x ) d x$ , or $f ' ( x )$ .

In fact if $d f$ and $d x$ are differentials, then the derivative is $d f d x$ by definition since the derivative is the ratio of the change of $f$ to the change of $x$ in the linear approximation.

The applet here allows you to enter any standard function and domain, and look at it, its slope and derivative in it.