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:

Δ y Δ x = f ( x + h ) - f ( x ) x + h - x = f ( x + h ) - f ( x ) h

When you divide by h (met h 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 ( d y ) ( dx ) for any given x.

The function is accordingly called the x derivative function. You write it as f ' ( x ) .

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 ' ( x ) is the graph of the slopes of f .

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