Given
[tex]w = xe^{y/z},\ x = t^7,\ y = 4 - t, \ z = 2 + 9t \\ \\ \frac{dw}{dt} = \frac{dw}{dx} \cdot \frac{dx}{dt} + \frac{dw}{dy} \cdot \frac{dy}{dt} + \frac{dw}{dz} \cdot \frac{dz}{dt} \\ \\ \frac{dw}{dx}=e^{y/z} \\ \\ \frac{dw}{dy}= \frac{x}{z} e^{y/z} \\ \\ \frac{dw}{dz}=- \frac{xy}{z^2} e^{y/z} \\ \\ \frac{dx}{dt}=7t^6 \\ \\ \frac{dy}{dt}=-1 \\ \\ \frac{dz}{dt}=9[/tex]
Thus,
[tex] \frac{dw}{dt}=e^{y/z}\cdot7t^6+\frac{x}{z} e^{y/z}\cdot(-1)+- \frac{xy}{z^2} e^{y/z}\cdot(9) \\ \\ =7t^6e^{y/z}-\frac{x}{z} e^{y/z}-9\frac{xy}{z^2} e^{y/z} \\ \\ =\left(7t^6-\frac{x}{z}-9\frac{xy}{z^2}\right)e^{y/z}[/tex]
