Derivative of a multivariable composite functions

Question

Let $\;f(x,y,z)=xz+xy+yz\;$ and $\;g(t)=(e^t,\cos t,\sin t)$. The thing is that I want to calculate $(f\circ g)'(1)$. Previously, I was asked to calculate $g'$ and $\nabla f$, so I do not know if that is necessary to calculate $(f\circ g)'(1)$ or I can simply composite the functions and apply chain rule for $t$.

Answer 1

The point is to do it by the multivariable chain rule, not by single-variable calculus. Indeed, you should understand that if I tell you $\nabla f(a,b,c)$ and tell you that $g(1)=(a,b,c)$, then knowing only the vector $g'(1)$ and not any more, you can find $(f\circ g)'(1)$. Namely, it is $$(f\circ g)'(1) = \nabla f(g(1))\cdot g'(1).$$ Again, I emphasize that you only need to know the velocity vector at the point and not the actual curve $g(t)$ in order to find $(f\circ g)'(t_0)$.