Solution
= Solution
The <multivariable chain rule> gives
$$
\frac{dF}{dt}=(3x^2yz+y)(-3\sin t)+(x^3z+x)(3\cos t)+(x^3y+1)2.
$$
Substituting the three parametrizing functions also writes the complete <derivative> as
$$
\frac{dF}{dt}=162\cos^3t\sin t
+162t(\cos^4t-3\cos^2t\sin^2t)+9\cos2t+2.
$$
At $t=\pi/2$, the point is $(0,3,\pi)$, its velocity is $(-3,0,2)$, and $\nabla F=(3,0,1)$. Hence
$$
\boxed{\left.\frac{dF}{dt}\right|_{t=\pi/2}=\nabla F\cdot\mathbf r'=-9+2=-7}.
$$