equation -- Wave Equation Derivation Question

Hello, my teacher says that if, on a wave equation

[tex]f(x-ct)=f(e)[/tex] then
[tex]\partial_{ee}= \partial_{tt}- c^2 \partial_{xx}[/tex]

but i think that

[tex]\partial_{t}=\frac{\partial }{\partial e} \frac{\partial e}{\partial t}=-c\frac{\partial }{\partial e} [/tex]

and

[tex]\partial_{x}=\frac{\partial }{\partial e} \frac{\partial e}{\partial x}=\frac{\partial }{\partial e} [/tex]

then

[tex]\partial_{tt}- c^2 \partial_{xx}= c^2 \partial_{ee}- c^2 \partial_{ee}=0[/tex]

what is the correct?