Let $x(t)\in R^n$ be a time-dependent variable and consider two vector-valued functions $g : R^n \mapsto R^m$ and $f : R^m \mapsto R^p$. What is the time derivative $\frac{d}{dt}f(g(x(t)))$ ?
As I understand, applying the chain rule we have that
\begin{equation} \frac{d}{dt}f(g(x(t))) = \frac{\partial }{\partial x} f(g(x)) \frac{dx}{dt}. \end{equation}
The partial derivative (a Jacobian matrix, actually) is given by
\begin{equation} \frac{\partial }{\partial x} f(g(x)) = D_f(g(x)) D_g(x), \end{equation}
where $D_f(a)$ is the Jacobian matrix of a function $f$ evaluated at $a$.
Is this correct? Or should it be $(D_f(g(x)))^T$ in the equation above?
$$\frac{d}{dt}(f(g(x(t))))=J_f(g(x(t)))J_g(x(t))\begin{bmatrix}\dot{x_1}\\.\\.\\.\\\dot{x_n}\end{bmatrix}$$