`f(x_0+h)=4*(x_0+h)^3+5`
`f(x_0)=4*x_0^3+5`
`(f(x_0+h)-f(x_0))/(h)=(4*(x_0+h)^3+5-(4*x_0^3+5))/h=4*((x_0+h)^3-x_0^3)/h`
Bekannt ist: `(a+b)^3=a^3+3*a^2*b+3*a*b^2+b^3`
`(f(x_0+h)-f(x_0))/(h)=4*(x_0^3+3*x_0^2*h+3*x_0*h^2+h^3-x_0^3)/h`
`=4*(3*x_0^2*h+3*x_0*h^2+h^3)/h=4*(3*x_0^2+3x_0*h+h^2)`
`lim_(hto0)(f(x_0+h)-f(x_0))/(h)=lim_(hto0)4*(3*x_0^2+3x_0*h+h^3)=12*x_0^2`
Student, Punkte: 5.08K