So I'm working with Jacobimatrices. However, the lecture notes claim two results which I cannot figure out since there are no explanation of it.
So let $\textbf{F}:\mathbb{R}^2\rightarrow \mathbb{R}^2$ is a function given below $$\textbf{F}(x,y)=(f_1(x,y),f_2(x,y))$$ where we know that $f_1,f_2:\mathbb{R}^2\rightarrow \mathbb{R}$. We define the Jacobimatrix as follows, $$D\textbf{F}(x,y)=\begin{bmatrix} \frac{\partial}{\partial x}f_1(x,y) & \frac{\partial}{\partial y}f_1(x,y)\\ \frac{\partial}{\partial x}f_2(x,y) & \frac{\partial}{\partial y}f_2(x,y) \end{bmatrix}$$ Further, we define $c:\mathbb{R}^2\rightarrow \mathbb{R}^2$ as follows $$\textbf{c}(t)=(c_1(t),c_2(t))$$ Now here is the first claim: $$\frac{d}{dt}\textbf{F}(\textbf{c}(t))=D\textbf{F}(\textbf{c}(t))\textbf{c}'(t)$$
Before the second claim, we first define $\textbf{G}:\mathbb{R}^2\rightarrow \mathbb{R}^2$ is a function given below $$\textbf{G}(x,y)=(g_1(x,y),g_2(x,y))$$ Then we say $\textbf{H}(x,y)=\textbf{F}(\textbf{G}(x,y))$
The second claim are as follows: $$D\textbf{H}(x,y)=D\textbf{F}(\textbf{G}(x,y))D\textbf{G}(x,y)$$
I cannot see how this could be true. Can anyone explain these two claims for me? Thanks in advance