Change > Derivative values

Here you see part of the grph of the function $y=f\left(x\right)$ .

The mean change of the function $f$ on the interval $[a,b]$ is:

$\frac{(\Delta y)}{(\Delta x)}=\frac{(f\left(b\right)-f\left(a\right))}{(b-a)}$

The rate of change at the point $x=a$ can be found by calculating the difference quotient for the interval $[a,a+h]$ :

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

You continue to reduce $h$ until it approaches $0$ .

This gives you a sequence of difference quotients.

In this sequence, the value of the difference quotients will be approaching a certain
value.

This value is the derivative
$\frac{\left(\text{d}y\right)}{\left(\text{d}x\right)}$ at *x* = *a*.

It is the rate of change of the function $f$ at $x=a$ .

It is also the slope of the tangent at $x=a$ for the graph of $f$ .

You write: $f\text{'}\left(a\right)$ .

On the graphing calculator a derivative is written as dy/dx.