Change > Derivative values
12345Derivative values

## Theory

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 $\left[a,b\right]$ is:
$\frac{\left(\Delta y\right)}{\left(\Delta x\right)}=\frac{\left(f\left(b\right)-f\left(a\right)\right)}{\left(b-a\right)}$

The rate of change at the point $x=a$ can be found by calculating the difference quotient for the interval $\left[a,a+h\right]$ :
$\frac{\left(\Delta y\right)}{\left(\Delta x\right)}=\frac{\left(f\left(a+h\right)-f\left(a\right)\right)}{h}$
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.