Question

$$\sqrt{a^{2}z^{2}+1}; d_{z}$$

Evaluate

$\left\{\begin{matrix}\frac{\ln(|\sqrt{\left(az\right)^{2}+1}+z|a||)+z|a|\sqrt{\left(az\right)^{2}+1}}{2|a|}+С,&a\neq 0\\z+С,&a=0\end{matrix}\right.$

Differentiate w.r.t. a

$\left\{\begin{matrix}\frac{\left(\sqrt{\left(az\right)^{2}+1}-z|a|\right)\left(-\left(az\right)^{2}\ln(|\sqrt{\left(az\right)^{2}+1}+z|a||)-z|a|\sqrt{\left(az\right)^{2}+1}\ln(|\sqrt{\left(az\right)^{2}+1}+z|a||)-\ln(|\sqrt{\left(az\right)^{2}+1}+z|a||)+\left(az\right)^{4}+|a|a^{2}z^{3}\sqrt{\left(az\right)^{2}+1}+azsign(a)\sqrt{\left(az\right)^{2}+1}+\left(az\right)^{2}\right)}{2a|a|\sqrt{\left(az\right)^{2}+1}},&a\neq 0\\0,&a=0\end{matrix}\right.$