# Derivative of Bessel function at x=0.

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Selçuk Sehitoglu on 15 May 2022
Commented: Star Strider on 21 May 2022
Hello everyone,
I would like to change the expression of derivative of besselj function, because embedded expression is leading some problems.
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) in MATLAB and it is undefined at x=0.
However, the answer is available with the following expression (mathematically):
-(besselj(n-1,x) - besselj(n+1,x))/2
Is there a way to use this expression?
Ozi

Star Strider on 15 May 2022
One option is to use the limit function, however tha does not always produce a numeric result.
I ususally cheat and use eps instead.
syms nu x
y(x) = besselj(nu,x);
d_y = diff(y,x)
d_y(x) = d_y0 = limit(d_y,x,0)
d_y0 = d_y0 = d_y(eps) % Cheat With 'eps'
d_y0 = d_y0 = vpa(d_y0)
d_y0 = Since ν is undefined, a double result is not possible.
.
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Star Strider on 21 May 2022
As always, my pleasure!