Differentiation of a composition of functions in different variables

Question

Given $c:\mathbb{R} \rightarrow \mathbb{R^{3}}$; $c(t)=(t_{1},t_{2}, t_{3})$ and $f:\mathbb{R^{3}} \rightarrow \mathbb{R^{3}}$ ; $f(x,y,z)=(x_{1},y_{1},z_{1})$ how can I compute $(f\circ c)'(0)$. Is there a way to use a modified version of the chain rule? where $c(0)= \bf{p} \in \mathbb{R^3}$ and $c'(0)=X \in \mathbb{R^3}$. Apparently I should arrive at $f(X)$.

Answer 1

Chain rule still applies as long as you use total derivatives, i.e. the gradient (or in this context, the Jacobian):

$$\frac{d}{dt}f(c(t)) = \nabla f (c(t)) \cdot \frac{dc}{dt}(t)$$

Answer 2

The definition of your functions do not make a lot of sense (e.g. what are $t_1$, $x_1$ etc.?). Anyways, the chain rule still applies: $(f \circ c)'(0) = f'(c(0)) \cdot c'(0)$, where $f'$ is the Jacobi matrix in this case.