[snip]

the solution of the heat equation $\displaystyle u_{t} = u_{xx}$ with initial condition $\displaystyle u(x,0) = f(x)$ can be written as

$\displaystyle u(x,t) = \int_{-\infty}^{\infty} \frac{1}{2\sqrt{{\pi}t}}\exp\left(-\frac{(x-x')^{2}}{4t}\right)f(x') dx'$

[snip]

The next bit is the problem:

c) By taking the singular data for the heat equation $\displaystyle f(x) = \delta^{\left(n\right)}(x)$ construct a solution $\displaystyle u(x,t)$ of the heat equation.

Now, I've gone through the solutions and the line has jumped from:

$\displaystyle u(x,t) = \frac{1}{2\sqrt{{\pi}t}}\int_{-\infty}^{\infty} \exp\left(-\frac{(x-x')^{2}}{4t}\right)f(x') dx'$

to

$\displaystyle u(x,t) = \frac{1}{2\sqrt{{\pi}t}}\left(-\frac{\partial}{{\partial}x}\right)^{n}\exp\left(-\frac{x^{2}}{4t}\right)$

This is where I'm stuck. I'm guessing it's some kind of delta function property but I'm not sure. The $\displaystyle \delta^{\left(n\right)}$ threw me a little. If someone can explain what's going on, that'd be great!

Thanks.