$u=x^2+y^2+z^2\\ x=e^t, y=e^tsint,z^t cost\\ \dfrac{\partial u}{\partial x}=2x, \dfrac{\partial u}{\partial y}=2y, \dfrac{\partial u}{\partial z}=2z, ---(1)\\ \dfrac{\partial x}{\partial t}=e^t, \dfrac{\partial x}{\partial t}=e^t sint+e^tcost, \dfrac{\partial x}{\partial t}=e^t cost-e^tsint,---(2) \ \\ \dfrac{du}{dt}=\dfrac{\partial u}{\partial x}.\dfrac{\partial x}{\partial t}+\dfrac{\partial u}{\partial y}.\dfrac{\partial y}{\partial t}+\dfrac{\partial u}{\partial z}.\dfrac{\partial z}{\partial t} \\ Substituting \ the \ value\ for\ equation (1) and (2)$

$\dfrac{\partial u}{\partial t}=2x.e^t+2y(e^tsint+e^tcost)+2y(e^tcost-e^tsint) \\ =2e^t.e^t+2e^t.sint(e^tsint+e^tcost)+2e^t.cost(e^tcost-e^tsint) \\ =2e^{2t}+2e^{2t}( sin^2t+cos^2t)+2e^{2t}(sint.cost-sint.cost)\\ =2e^{2t}+2e^{2t}(1)\\ =4e^{2t}----- Proved$