Uniform first-coordinate bound for the derivative of the Dickman covariance kernel

Let $\rho=\rho_5$ be the finite method-of-steps Dickman function, let $U=9/2$, and put $F(x)=\rho(U-x)/\rho(U)$. There is a real constant $C>0$ such that, for every pair of real numbers $s,t\in[0,1]$,

The derivative is the ordinary real derivative, represented by Lean's deriv. The estimate is uniform over the whole unit square, including the boundary $s=0$. It controls the derivative in the second variable of the normalized Dickman covariance kernel and its vanishing at the first-coordinate boundary.