# Symbolic differentiation of Bessel functions is incorrect

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Sam Spedding on 11 Mar 2021
Commented: David Goodmanson on 16 May 2022
The output of the below code gives an incorrect equation for the derivative of the modified Bessel function of the second kind.
syms x n
y = besselk(n,x);
diff(y,x)
It says the derivative of is
but as I understand, the formula for the derivative of the K bessel functions is given by
.
What's going on?!?

David Goodmanson on 11 Mar 2021
Edited: David Goodmanson on 11 Mar 2021
Hi Sam,
There is really nothing going on. Both of those identities are correct, as you can check numerically. There are several recurrence relations available for Bessel functions. Another one is
K'(n,x) = -nK(n,x)/x - K(n-1,x)

Selçuk Sehitoglu on 15 May 2022
Hello,
i am dealing with the same problem for besselj.
syms nu x
y=besselj(nu,x);
d_y=diff(y,x);
subs(d_y,x,0); % At this row i get "Division by zero." error, because derivative is defined as -nJ(n,x)/x - J(n-1,x) ans it is undefined at x=0.
However, the answer is available with the following expression:
-(J(n-1,x) - J(n+1,x))/2
Is there a way to use this expression?
Ozi
David Goodmanson on 16 May 2022
Hi Ozi,
Yes there is. The expression works as is. It looks like n = 0 might be an exception because you will need J(-1,x) but for real n the besselj function works for negative n.