Finding the nth Derivative of Cosine Function

[matematikuvol]

Homework Statement

[tex]\frac{d^{2n}}{dx^{2n}}\cos x[/tex]

[tex]n \in N[/tex]

Homework Equations

[tex]\cos x=\sum^{\infty}_{k=0}(-1)^k\frac{x^{2k}}{(2k)!}[/tex]

The Attempt at a Solution

[tex]\frac{d^{2n}}{dx^{2n}}x^{2n}=(2n)![/tex]

But k is different that [tex]n[/tex]. I don't have a clue how to solve that.

 

[Char. Limit]

Here's a recommendation: Check the derivative for certain small values of n and see if you can find a pattern. I recommend n=0, n=1, and n=2. Then just remember that

[tex]\frac{d^{n+4}}{d x^{n+4}} cos(x) = \frac{d^n}{d x^n} cos(x)[/tex]

And that should finish your problem.