Please could someone suggest how do i prove that d^(2n)y/dx(2n) =(-1(^n))*sin(x)
basically if you keep finding the first, second third etc. differential of sinx or cosx they always continue in a cycle how do i formally prove ty
I recommend Induction. Check out the Wikipedia article on how to word a proof by induction:
https://en.wikipedia.org/wiki/Mathematical_induction
I am no fan of proofs by induction for cyclic formulas
$ \begin{align*}y&=\sin(x)\\y^{(1)}&=\cos(x)\\y^{(2) }&=-\sin(x)\\y^{(3)}&=-\cos(x)\\y^{(4)}&=\sin(x) \end{align*}$
We now have a complete cycle.
Suppose we want the $146^{th}$ derivative. Well $146=4(36)+2$ hence $y^{(146)}=-\sin(x)$