= Solution
The <derivative> formula requires additional regularity: continuity alone does not define $f'$. Assume $f$ is continuously differentiable, or locally absolutely continuous with its almost-everywhere <derivative>. Integration by parts on $[0,R]$ gives
$$
\int_0^Re^{-pt}f'(t)dt=e^{-pR}f(R)-f(0)+p\int_0^Re^{-pt}f(t)dt.
$$
For $\operatorname{Re}p>c$, the given exponential bound makes the first term tend to zero. Thus the improper transform exists under these regularity assumptions and
$$
\boxed{\widehat{f'}(p)=p\widehat f(p)-f(0).}
$$
For the primitive, absolute integrability justifies reversing the integration order:
$$
\widehat h(p)=\int_0^\infty f(s)\int_s^\infty e^{-pt}dt\,ds
=\boxed{\frac{\widehat f(p)}p}.
$$
Indeed the corresponding absolute double integral is finite when $\operatorname{Re}p>c$.
For real $p>0$, substitution $t=s^2/p$ and the supplied <Gaussian integral> give
$$
\mathcal L[t^{-1/2}](p)=\frac2{\sqrt p}\int_0^\infty e^{-s^2}ds
=\boxed{\sqrt\pi\,p^{-1/2}.}
$$
Both sides are <holomorphic> for $\operatorname{Re}p>0$, so the <identity theorem> extends this equality there, taking the square root with positive real part. The integrable singularity at zero causes no difficulty for this improper integral. The primitive rule then gives $\mathcal L[2\sqrt t]=\sqrt\pi\,p^{-3/2}$.
For the conductor, take the <Laplace transform> in time and write $U(x,p)$. The initial temperature gives $U_{xx}=pU$. Use the usual physical condition that the temperature is bounded far down the conductor; this selects the decaying solution
$$
U(x,p)=A(p)e^{-\sqrt p\,x},\qquad\operatorname{Re}\sqrt p>0.
$$
The constant applied flux has transform $Q/p$, so $-kU_x(0,p)=Q/p$, giving $A(p)=Q/(kp^{3/2})$. This is the <surface temperature under constant heat flux on a half-line>. Using the transform just calculated yields
$$
\boxed{u(0,t)=\frac{2Q}{k\sqrt\pi}\sqrt t\qquad(t>0).}
$$
Here $k>0$ is the <thermal conductivity>. The far-field boundedness is the customary physical selection condition; without a spatial growth restriction the written PDE and initial/boundary data alone do not specify that selection. Within the physical solution, the face temperature tends continuously to zero as $t\downarrow0$.
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