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.
All norms on $\mathbb{R}^n$ are equivalent, meaning for any two norms $\|\cdot\|_1, \|\cdot\|_2$, there exist constants $A,B$ such that $A\|x\|_2 \leq \|x\|_1 \leq B\|x\|_2$. Since you can view a matrix as a vector in $\mathbb{R}^{n^2}$, this applies to matrix norms too. You haven't specified which matrix norm you're using, but for each norm, there will be a $C$ such that
$$\|M\|\leq C\max_{i,j}|M_{ij}|$$
For any matrix $M$. So your stated bound indeed holds, but the constant will depend on the norm you use (see the end of the Wikipedia article on matrix norms for precise constants, which are often sharp).