Our purpose in this section is to solve the problem $$ \begin{aligned} u_{t} &=k u_{x x} \quad(-\infty<x<\infty, 0<t<\infty) \\ u(x, 0) &=\phi(x) . \end{aligned} $$
We'll look for $Q(x, t)$ of the special form $$ Q(x, t)=g(p) \quad \text { where } p=\frac{x}{\sqrt{4 k t}} $$
$$Q_{t}=\frac{d g}{d p} \frac{\partial p}{\partial t}=-\frac{1}{2 t} \frac{x}{\sqrt{4 k t}} g^{\prime}(p)$$
Wolframalpha says that $$Q_{t}=-\frac{x g^{\prime}\left(\frac{x}{2 \sqrt{k t}}\right)}{4 \sqrt{k} t^{3 / 2}}$$
I think this is in contradiction to the reference calculation and I see no way to obtain the reference calculation.
Who is correct?