Let γ : [ 0 , 2 π ] → R 3 be γ (...

Step 1
A vector valued function $f:{\mathbb{R}}^n\to {\mathbb{R}}^m$ is differentiable if its component functions $f_i:{\mathbb{R}}^n\to \mathbb{R}$ are differentiable. Here, $n=1$ and $m=3$ The component functions are,
$$\begin{gathered} f_1(t)=4t \\ {f}_2(t)=\cos (3t) \\ {f}_3(t)=\sin (3t) \end{gathered}$$
Hopefully it is clear that each of these component functions are differentiable. Then, we have that f is differentiable and its derivative is the Jacobian (matrix of partial derivatives),
$\mathrm{\nabla }f=\left(\begin{array}{c}{\mathrm{\partial }}_t{f}_1\\ {\mathrm{\partial }}_t{f}_2\\ {\mathrm{\partial }}_t{f}_3\end{array}\right)=\left(\begin{array}{c}4\\ -3\sin (3t)\\ 3\cos (3t)\end{array}\right)$