[Math] How to differentiate a vector with respect to time

Question

Say I have: $\vec{A}(\vec{x}(t))$ and I want to find $\frac{d\vec{A}}{dt}$. Can someone please tell me how to do this? I am getting confused regarding the chain rule here because we can write out $\vec{A} = <A_x,A_y,A_z>$ and each of those is a function of time.

Answer 1

If

$$\mathbf{A} = A(\mathbf{x}(t))$$

Then

$$\frac{\text{d}\mathbf{A}}{\text{d}t} = \frac{\text{d}A}{\text{d}x}\frac{\text{d}x}{\text{d}t}$$

Now just extend the rule with the chain method for more than the single $x$ component...