Solved – chain rule in differentiating a CDF

You cannot create a CDF simply by taking a known valid CDF $F_Z(z)$ and replacing its argument $z$ by $sqrt{z}$. Remember that a CDF is a real-valued function defined for all values of its argument, and when you write $F_Z(sqrt{z})$, you are not telling us what value your new alleged CDF has when the argument of your new CDF has value smaller than $0$. Now, if $Z$ is a nonnegative random variable, then the function $$G(z) = begin{cases} F_Z(sqrt{z}), & z geq 0,\0, & z < 0,end{cases}tag{1}$$ is indeed a valid CDF of a different random variable $Y$ (it happens to equal $Z^2$) and if $F_Z(z)$ happens to be an absolutely continuous function, then the derivative of $G(z)= F_Y(z)$ which is $$frac{mathrm d}{mathrm dz}G(z) = g(z) = begin{cases} f_Z(sqrt{z})frac{1}{2sqrt{z}}, & z geq 0,\0, & z < 0,end{cases}$$ is the pdf of this different random variable.