Finding the nth Derivative of Cosine Function

[matematikuvol]

Homework Statement

$$\frac{d^{2n}}{dx^{2n}}\cos x$$

$$n \in N$$

Homework Equations

$$\cos x=\sum^{\infty}_{k=0}(-1)^k\frac{x^{2k}}{(2k)!}$$

The Attempt at a Solution

$$\frac{d^{2n}}{dx^{2n}}x^{2n}=(2n)!$$

But k is different that $$n$$. 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

$$\frac{d^{n+4}}{d x^{n+4}} cos(x) = \frac{d^n}{d x^n} cos(x)$$

And that should finish your problem.