f(x, y_1, y_2, y_3) = (y_1 \langle a, x \rangle + b_1 \cos\|x\|^2, y_2 \langle a, x \rangle + b_2 \cos\|x\|^2, y_3 \langle a, x \rangle + b_3 \cos\|x\|^2) \\ 

\implies \\

\partial_{x_i} f_j (x,y) = y_j a_i - 2b_j \sin \|x\|^2 x_i \\

\partial_{y_i} f_j (x,y) = \langle a, x \rangle