Derivative functions > Derivative

You can approximate the slope of the graph of a function
$f$ for a given point value of *x* using the difference quotient over the interval
$[x,x+h]$ . You let
$h$ get closer and closer to
$0$ and check if the difference quotient is approaching a certain limit value. If this
is the case, then you have found the derivative, the required slope. The difference
quotient is defined as:

$\frac{\Delta y}{\Delta x}=\frac{f(x+h)-f\left(x\right)}{x+h-x}=\frac{f(x+h)-f\left(x\right)}{h}$

When you divide by
$h$ (met
$h\ne 0$ ) you are left with an expression that only depends on the value of
$x$ as
$h$ gets closer to
$0$ . (Although the slope in the above graph is positive,
$h$ can of course also be negative!)

This is the derivative
$\frac{\left(\text{d}y\right)}{\left(\text{dx}\right)}$ for any given *x*.

The function is accordingly called the $x$ derivative function. You write it as $f\text{'}\left(x\right)$ .

This derivative represents the slope of the graph of the function $f$ for any given $x$ . It is also the slope of the tangent of the graph of $f$ at this value of $x$ .

The graph of $f\text{'}\left(x\right)$ is the graph of the slopes of $f$ .