Solving Partial Derivative Equation: Finding Error & Fixing It

[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
$$(bx-ct)^2$$