f ( x ) = x^3
2 Punkte der Funktion
P ( x | x^3 )
P ( x + h | ( x + h ) ^3 )
Steigung zwischen den Punkten
Differenzenquotient
m = Δ y / Δ x = ( y1 - y2 ) / ( x1 - x2 )
m = [ x^3 - ( x + h )^3 ] / [ x - ( x + h ) ]
m = [ x^3 - ( x^3 + 3*h*x^2 + 3*h^{2} *x + h^3 ) ] / ( - h )
m = ( x^3 - x^3 - 3*h*x^2 - 3*h^2 *x - h^3 ) / ( - h )
m = ( - 3*h*x^2 - 3*h^2*x - h^3 ) / ( - h )
m = ( - h ) * ( 3*x^2 + 3*h*x + h^2 ) / ( - h )
m = 3 * x^2 + 3 *h*x + h^2
Differentialquotient
lim h −> 0 [ 3 * x^2 + 3 *h * x + h^2 ] =
f ´( x ) = 3 * x^2