Q1: Let $f,g: \mathbb{R}^n \rightarrow \mathbb{R}^m$ be two totally differentiable functions. What is correct for the function $f + 3 g$?

A1: $d(f + 3 g)_x = df_x + 3 dg_x$

A2: $d(f + 3 g)_x = df_x - 3 dg_x$

A3: $d(f + 3 g)_x = df_x + dg_x$

A4: $d(f + 3 g)_x = 3 df_x + dg_x$

A5: $d(f + 3 g)_x = 3 (df_x + dg_x)$

Q2: Let $g: \mathbb{R}^k \rightarrow \mathbb{R}^n$ and $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$ be two totally differentiable functions. What is not correct for the composition $f \circ g$?

A1: $d(f \circ g)x = df{x} + dg_x$

A2: $d(f \circ g)x = df{g(x)} \circ dg_x$

A3: $J_{f \circ g}(x) = J_{f}(g(x)) J_{g}(x)$

A4: $(f \circ g)(x) = f(g(x))$

Q3: Let $g: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ and $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be two totally differentiable functions given by $$ g(x_1,x_2) = \begin{pmatrix} 2 +x_2 \ x_1 \end{pmatrix}$$ and $$ f(x_1,x_2) = \begin{pmatrix} x_1^2 \ 3 \end{pmatrix}$$ What is $J_{f \circ g}(x)$?

A1: $\begin{pmatrix} 0 & 4 + 2 x_2 \ 0 & 0 \end{pmatrix}$

A2: $\begin{pmatrix} 1 & 4 + x_2 \ 0 & 1 \end{pmatrix}$

A3: $\begin{pmatrix} 0 & 4 + 4 x_2 \ 1 & 0 \end{pmatrix}$

A4: $\begin{pmatrix} 0 & 4\ 1 & 0 \end{pmatrix}$