Partial Derivative

[dimensionless]

I'm trying to figure out this equation.


{\Psi} = Ae^{-a(bx-ct)^2}


I've expanded this to


{\Psi} = Ae^{-ab^2x^2-abxct-ac^2t^2}


When I try to find the derivative I get this


\left(\newcommand {\pd}[3]{ \frac{ \partial^{#3}{#1} }{ \partial {#2}^{#3} } }

\pd{\Psi}{t}{}\right)_x = (-2ac^2t-abxc)Ae^{-ab^2x^2-abxct-ac^2t^2}


I should get this instead


\left(\newcommand {\pd}[3]{ \frac{ \partial^{#3}{#1} }{ \partial {#2}^{#3} } }

\pd{\Psi}{t}{}\right)_x = (-2abcx-2ac^2t)Ae^{-ab^2x^2-abxct-ac^2t^2}


Can anyone tell me where my error is and how I can fix it?

[krab]

Try again in expanding
(bx-ct)^2