I dont know where I am gone wrong. $\displaystyle u(x,t)= \int^{\infty}_{-\infty} \frac{e^{-(x-\xi)^2/4kt}}{2\sqrt {\pi kt}} f(\xi)d\xi$

Using the chain rule I calculate for $\displaystyle u_x, u_x_x$

$\displaystyle u_x=\frac{\frac{-2(x-\xi)e^{-(x-\xi)^2/4kt}}{4kt}}{2\sqrt {\pi k t}}$ and

$\displaystyle u_x_x=\frac{\frac{4(x-\xi)e^{-(x-\xi)^2/4kt}}{4kt}}{2\sqrt {\pi k t}}$

and the product rule for $\displaystyle u_t$ since t is on both numerator and denominator

$\displaystyle \frac {2\sqrt {\pi k t} (\frac{(x-\xi)^2}{4kt^2}e^{-(x-\xi)^2/4kt}) - (e^{-(x-\xi)^2/4kt} {(\pi k )}{\pi k t}^{-1/2})}{4\pi kt}$

Can anyone confirm im going correct so far?