The Chain Rule & Implicit Differentiation · Superchallenge problem

A function of the form $u(x,t) = \varphi\!\left(\dfrac{x}{\sqrt t}\right)$, with $\varphi$ twice differentiable, solves the heat equation $u_t = u_{xx}$ for all $x$ and all $t > 0$. If $\varphi'(0) = 1$, find $u_x(4, 4)$.