Change > Derivative values
12345Derivative values

Theory

Here you see part of the grph of the function y = f ( x ) .

The mean change of the function f on the interval [ a , b ] is:
( Δ y ) ( Δ x ) = ( f ( b ) - f ( a ) ) ( 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 ] :
( Δ y ) ( Δ x ) = ( f ( a + h ) - f ( a ) ) 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 ( d y ) ( d x ) 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 ' ( a ) .

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

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