Hello,
it seems that the derivatives of bessel_K have the wrong sign:

sage: [Bessel(i, 'K')(x).diff(x) for i in range(5)]
[1/2*bessel_K(1, x) + 1/2*bessel_K(-1, x), 1/2*bessel_K(2, x) + 
1/2*bessel_K(0, x), 1/2*bessel_K(3, x) + 1/2*bessel_K(1, x), 
1/2*bessel_K(4, x) + 1/2*bessel_K(2, x), 1/2*bessel_K(5, x) + 
1/2*bessel_K(3, x)]

on:
Sage Version 6.1.1, Release Date: 2014-02-04
CPU: AMD Athlon(tm) 64 X2 Dual Core Processor 4200+
OS: Ubuntu 12.04.5 LTS

For type J, Y and I Bessel functions the derivatives appear to be right. 

Best regards,
Roman. 

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