Derivative of a sine function

[Aion]

A derivative of ordern [MATH]n[/MATH] can be defined inductivley as [MATH]D^{n} f=D(D^{n-1}f)[/MATH].

Given that [MATH]D sin(x) = cos(x)=sin(x+\pi/2)[/MATH]
we obtain, according to my mathbook, [MATH]D^{n} sin(x)=Sin(x+n\pi/2)[/MATH].

But how can one prove this formula?

 

[lev888]

A derivative of ordern [MATH]n[/MATH] can be defined inductivley as [MATH]D^{n} f=D(D^{n-1}f)[/MATH].

Given that [MATH]D sin(x) = cos(x)=sin(x+\pi/2)[/MATH]
we obtain, according to my mathbook, [MATH]D^{n} sin(x)=Sin(x+n\pi/2)[/MATH].

But how can one prove this formula?

One derivative adds one [MATH]\pi/2[/MATH]. n derivatives add how many?

 

[HallsofIvy]

Since it depends upon the positive integer, n, use induction on n.