Assume the temperature of the exhaust in an exhaust pipe can be approximated by \(T=T_{0}\left(1+a e^{-b x}\right)[1+c \cos (\omega t)]\), where \(T_{0}=100^{\circ} \mathrm{C}, a=3, b=0.03 \mathrm{~m}^{-1}, c=0.05\), and \(\omega=100 \mathrm{rad} / \mathrm{s}\). If the exhaust speed is a constant \(3 \mathrm{~m} / \mathrm{s}\), determine the time rate of change of temperature of the fluid particles at \(x=0\) and \(x=4 \mathrm{~m}\) when \(t=0\).