Calculating differential equation with bessel function in it with ln How can I solve that:$$\frac{d}{dx}\ln \left(\frac{xJ_{-\frac14}\left(\frac{x^2}{2}\right)}{2} \right)$$without using the fact$$2J^{'}_{\nu}(z)=J_{\nu-1}(z)-J_{\nu+1}(z)$$only the definition of the Bessel function
 A: Hint: 
The derivative of a Bessel function satisfies
\begin{eqnarray*}
2J^{'}_{\nu}(z)=J_{\nu-1}(z)-J_{\nu+1}(z)
\end{eqnarray*}
where differentiation is wrt to $z$.
You will still need to differentiate a product & some function of a function ... but thats easy ?
A: Another Hint:
$$\beta(x)= \frac{d}{dx}\ln \left(\frac{xJ_{-\frac14}\left(\frac{x^2}{2}\right)}{2} \right)$$
$$\beta(x)=\frac{d}{dx}\ln \left({J_{-\frac14}\left(\frac{x^2}{2}\right)} \right)+\frac {d}{dx} \left ( \ln \frac x2 \right ) $$
Then use for the first derivative your question here :
Calculating the integral of some bessel function
$$\int \frac {x\left(J_{ \frac 34}(\frac {x^2}{2})-J_{- \frac 54}(\frac {x^2}{2})\right)}{2J_{- \frac 14}(\frac {x^2}{2})}dx=-\ln \left ( J_{- \frac 14}(\frac {x^2}{2}) \right )$$
Take derivative on both sides
$$ \frac {x\left(J_{ \frac 34}(\frac {x^2}{2})-J_{- \frac 54}(\frac {x^2}{2})\right)}{2J_{- \frac 14}(\frac {x^2}{2})}=-\frac {d}{dx}\left (\ln \left ( J_{- \frac 14}(\frac {x^2}{2}) \right ) \right )$$
