Oh gosh, this one seems tricky, but really isn't. We just have to find a pattern. First of all, we know that the coefficient will be 2^50, but we don't exactly know the sign in front of it. If you do a pattern of derivatives:
0: cos(2x)
1: -2sin(2x)
2: -4cos(2x)
3: 8sin(2x)
4: 16cos(2x)
We'll find that every 4 derivatives, it returns to positive coefficient cosine. Therefore, if we fast forward to the 48th derivative, which is the closest multiple of 4 to 50, we'll find:
2^48cos(2x)
49th derivative: -2^49sin(2x)
50th derivative: -2^50cos(2x)
=-1125899906842624cos(2x)
That is the 50th derivative of cos(2x). Wow!
Hope that helped,
~Cam943
