Let $F:\mathbb{R}^3\setminus\{0\}\to\mathbb{R}^3$ be a function of class $C^1$ such that $$(x_1, x_2, x_3)\in \mathbb{R}^3\setminus\{0\}\mapsto F_i(x_1, x_2, x_3)\in\mathbb{R}, \quad \forall i\in\{1, 2, 3\}.$$
As an exercise, I have to find an upper bound (better if as tight as possible) for $$\left\Vert\left(\frac{\partial F_i}{\partial x_j}(y)\right)_{i, j=1, 2, 3}\right\Vert $$ where the above notation refers to the norm of the matrix $\left(\frac{\partial F_i}{\partial x_j}(y)\right)_{i, j=1, 2, 3}$ as it is defined in https://en.wikipedia.org/wiki/Matrix_norm (the norm defined through the supremum).
$\textbf{EDIT:}$ By following the idea given in the answer by @kieransquared, I checked again all the computations. The only information I have is that $$ \frac{\partial F_i}{\partial x_j}(y) \le\begin{cases} \displaystyle\frac{1}{\alpha^{\beta +1}} -\frac{y_i^2 (\beta +1)}{\gamma^{\beta +3}} &\hbox{ if } i=j\\[10pt] \displaystyle -\frac{y_j^2 (\beta +1)}{\gamma^{\beta +3}} &\hbox{ if } i\neq j, \end{cases}$$ where $a, \beta\ge 1$, $\alpha,\gamma >0$ constants and $|x|$ denotes the euclidean norm of the vector $(x_1, x_2, x_3)$.
Hence, by using the triangle inequality, it follows that $$ \left\vert\frac{\partial F_i}{\partial x_j}(y)\right\vert \le\begin{cases} \displaystyle\frac{1}{\alpha^{\beta +1}} +\frac{y_i^2 (\beta +1)}{\gamma^{\beta +3}} &\hbox{ if } i=j\\[10pt] \displaystyle \frac{y_j^2 (\beta +1)}{\gamma^{\beta +3}} &\hbox{ if } i\neq j, \end{cases}$$
Anyway, I can not deduce the desired upper bound from that information.
I hope someone could help. Thank you.