Problem 2. Let the pressure $p$ and temperature $T$ at a point $(x,y,z)$ be $$P(x,y,z)=\frac{x^2+2y^2}{1+z^2},\quad T(x,y,z)=5+xy-z^2$$
a. If the position of an airplane at time $t$ is $$(x(t),y(t),z(t))=\left(2t,t^2-1,\cos t\right)$$ find $\frac{\mathrm d}{\mathrm dt}(PT)^2$ at time $t=0$ as observed from the airplane.
Could anyone could walk me through this or even give me some pointers on how to start/proceed?
Use the formula for total derivative: $$ \frac{df}{dt} = \frac{\partial f}{\partial x}\frac{dx}{dt} + \frac{\partial f}{\partial y}\frac{dy}{dt}+\frac{\partial f}{\partial z}\frac{dz}{dt} $$