Solving Partial Derivative Equation: Finding Error & Fixing It

[dimensionless]

I'm trying to figure out this equation.

<br /> {\Psi} = Ae^{-a(bx-ct)^2}<br />

I've expanded this to

<br /> {\Psi} = Ae^{-ab^2x^2-abxct-ac^2t^2}<br />

When I try to find the derivative I get this

<br /> \left(\newcommand {\pd}[3]{ \frac{ \partial^{#3}{#1} }{ \partial {#2}^{#3} } }<br /> <br /> \pd{\Psi}{t}{}\right)_x = (-2ac^2t-abxc)Ae^{-ab^2x^2-abxct-ac^2t^2}<br />

I should get this instead

<br /> \left(\newcommand {\pd}[3]{ \frac{ \partial^{#3}{#1} }{ \partial {#2}^{#3} } }<br /> <br /> \pd{\Psi}{t}{}\right)_x = (-2abcx-2ac^2t)Ae^{-ab^2x^2-abxct-ac^2t^2}<br />

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

 

[krab]

Try again in expanding
(bx-ct)^2