Evaluate $\displaystyle \frac{dw}{dt}$ at t=2 for $\displaystyle w(x,y,z)=e^{xyz^{2}}$;$\displaystyle x=t$, $\displaystyle y=t$, $\displaystyle z=\frac{1}{t}$

From what I gather, I'm looking for:

$\displaystyle (\frac{\partial w}{\partial x}\frac{dx}{dt}+\frac{\partial w}{\partial y}\frac{dy}{dt}+\frac{\partial w}{\partial z}\frac{dz}{dt})e^{xyz^{2}}$

Is this even correct? If so, I've tried to go further and gotten:

$\displaystyle [(yz^{2})(yz^{2})+(xz^{2})(xz^{2})+(2xyz)(\frac{-2xy}{t})]e^{xyz^{2}}$

and then:

$\displaystyle [(y^{2}z^{4})+(x^{2}z^{4})-(\frac{4x^{2}y^{2}}{t})]e^{xyz^{2}}$

then I substitute :

$\displaystyle [(t^{2}(\frac{1}{t})^{4})+(t^{2}(\frac{1}{t})^{4})-(\frac{4t^{2}t^{2}}{t})]e^{(t)(t)(\frac{1}{t})^{2})}$

Then set t=2

$\displaystyle [\frac{4}{16}+\frac{4}{16}-(\frac{64}{2})]e^{1}=\frac{-63}{2}e$

I know that isn't the correct answer, so what am I doing wrong? Thanks.