I know it's almost not worth your time but I'm struggling way too much with this simple derivative: $\frac{d}{dt}[F(\gamma(t))]$
$F=F(x_1,...,x_n)=(F^1(x_1,...,x_n),...,F^n(x_1,...,x_n))$
$\gamma=\gamma(t)=(\gamma_1(t),...,\gamma_n(t))$, $t\in\Bbb R$
Notation: $\frac{\partial F^i(x)}{\partial x_j}\equiv F^i_{x_j}(x)$ and $\frac{d\gamma(t)}{dt}\equiv\gamma'(t)$
First let's calculate $\frac{d}{dt}F^i(\gamma(t))=\sum\limits_{j=1}^n F^i_{x_j}(\gamma_1(t),...,\gamma_n(t))\cdot\gamma_j'(t)=\langle \nabla F^i(\gamma(t)); \gamma'(t)\rangle$ where $\langle\ ;\ \rangle$ is the dot product.
Second $\frac{d}{dt}F(\gamma(t))=(\frac{d}{dt}F^1(\gamma(t)),...,\frac{d}{dt}F^n(\gamma(t)))=(\langle \nabla F^1(\gamma(t)); \gamma'(t)\rangle,...,\langle \nabla F^n(\gamma(t)); \gamma'(t)\rangle)$
Is that it?