Answer
$\dfrac{\partial u}{\partial \alpha} =4e^{-4} \\ \dfrac{\partial u}{\partial \beta} =-7e^{-4} \\ \dfrac{\partial u}{\partial \gamma} =-24e^{-4} $
Work Step by Step
$\dfrac{\partial u}{\partial \alpha} =e^{-4} \dfrac{\partial x}{\partial \alpha} -2e^{-4}\dfrac{\partial y}{\partial \alpha} +8e^{-4} \dfrac{\partial t}{\partial \alpha} $
Plug the values $\alpha=-1;\beta=2; \gamma=1$.
$\dfrac{\partial u}{\partial \alpha} =e^{-4} \dfrac{\partial x}{\partial \alpha} -2e^{-4}\dfrac{\partial y}{\partial \alpha} +8e^{-4} \dfrac{\partial t}{\partial \alpha} =4e^{-4} $
$\dfrac{\partial u}{\partial \beta} =(e^{-4}) \dfrac{\partial x}{\partial \alpha} -(2e^{-4}) \dfrac{\partial y}{\partial \beta} +(8e^{-4}) \dfrac{\partial t}{\partial \beta} $
Plug the values $\alpha=-1;\beta=2; \gamma=1$.
$\dfrac{\partial u}{\partial \beta} =(e^{-4}) \dfrac{\partial x}{\partial \alpha} -(2e^{-4}) \dfrac{\partial y}{\partial \beta} +(8e^{-4}) \dfrac{\partial t}{\partial \beta} =-7e^{-4} $
$\dfrac{\partial u}{\partial \gamma} =e^{-4} \dfrac{\partial x}{\partial \gamma} -2e^{-4}\dfrac{\partial y}{\partial \gamma} +8e^{-4} \dfrac{\partial t}{\partial \gamma} $
Plug the values $\alpha=-1;\beta=2; \gamma=1$.
$\dfrac{\partial u}{\partial \gamma} =(e^{-4}) \dfrac{\partial x}{\partial \gamma} -(2e^{-4}) \dfrac{\partial y}{\partial \gamma} +(8e^{-4}) \dfrac{\partial t}{\partial \gamma} =-24e^{-4} $
