
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 $x\text{'}$ in the interval other than $x$ itself, the difference between $\frac{f(x\text{'})f(x)}{x\text{'}x}$ 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 $\frac{df}{dx}$ on that line for any two point ${P}_{2}$ and ${P}_{1}$ on it, with ${P}_{1}=({P}_{1x},{P}_{1y})$ and ${P}_{2}=({P}_{2x},{P}_{2y})$
We use the notation $dx$ and $df$ 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 $\frac{df}{dx}$ or $\frac{df(x)}{dx}$ , or $f\text{'}(x)$ .
In fact if $df$ and $dx$ are differentials, then the derivative is $\frac{df}{dx}$ 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.
