Need help find the partial derivatives...I keep keeping cos(4x-3y+z)
Differentiate with respect to $x$. Note that $y$ is to be regarded as a constant but $z$ is not. So $$\bigl(\cos(-4x-3y+z)\bigr)\Bigl(-4+\frac{\partial z}{\partial x}\Bigr)=0\ .$$ See if you can take it from here.
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Differentiate with respect to $x$. Note that $y$ is to be regarded as a constant but $z$ is not. So $$\bigl(\cos(-4x-3y+z)\bigr)\Bigl(-4+\frac{\partial z}{\partial x}\Bigr)=0\ .$$ See if you can take it from here.