Doubt about how to find a Lipschitz constant I have a doubt about a sentence of my Calculus text. 

Let $f: [t_1, t_2]\times \mathbb{R}^n \to \mathbb{R}^n, (t,y)\to f(t,y)$ such that  $|\partial_{y_j} f_i|$ is continuous and bounded for every $i,j=1,...n$. Then f has $\sqrt n L$ as Lipschitz constant (with respect to y uniformly in t) where $L>0$ is such that $|\partial_{y_j} f_i|\le L$.

I don't know how to get $\sqrt n L$ as Lipschitz constant: for $i=1,...,n$ we have $|f_i(t,y)-f_i(t,z)|\le|\nabla f_i(t, \theta y)||y-z|$ for some $\theta\in[0,1]$; and since $|\nabla f_i(t, \theta y)|\le \sqrt{nL^2} $ we obtain $|\nabla f(t, \theta y)|\le \sqrt{n^2L^2}=nL$.
Do you know ways to improve my inequality? Thanks in advance.

I suspect it is using the fact that $\|x\|_2 \le \|x\|_1 \le \sqrt{n} \|x\|_2$.
Consider the path $x_0=x \to (y_1,x_2,...,x_n) \to (y_1,y_2,x_3,...,x_n) \to \cdots \to x_n=y$, where one component changes at a time.
\begin{eqnarray}
\|f(x,t)-f(y,t)\|_2 &\le& \sum_k \|f_(x_{k+1},t)-f(x_k,t)\|_2 \\
&\le& \sum_k L \|x_{k+1}-x_k\|_2 \\
&=& L \sum_k |y_k-x_k| \\
&=& L \|x-y\|_1 \\
&\le& \sqrt{n}L \|x-y\|_2
\end{eqnarray}
