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

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 ?
