Imagine that I have the following functions: $T,L:\mathbb{R}^{n} \longrightarrow \mathbb{R}^n$, such that: $T(\textbf{x})=(t_1(\textbf{x}),...,t_n(\textbf{x}))$ and $L(\textbf{x})=(l_1(\textbf{x}),...,l_n(\textbf{x}))$ with $\textbf{x}=(x_1,...,x_n)\in \mathbb{R}^n$.
Then we can write $(T \circ L)(\textbf{x})$ as:
$$(T \circ L)(\textbf{x})=(t_1(l_1(\textbf{x}),...,l_n(\textbf{x})),...,t_n(l_1(\textbf{x}),...,l_n(\textbf{x})))$$
Or, for short: $$(T \circ L)(\textbf{x})=(t_1(L(\textbf{x})),...,t_n(L(\textbf{x})))$$
With some examples I got the following equality:
$$\frac{\partial t_k(L(\textbf{x}))}{\partial l_i}=\frac{\partial t_k(\textbf{x})}{\partial x_i}\Bigg|_{\textbf{x}=L(\textbf{x})} \ \ \ \ \ k,i\in \{1,...,n\}$$
But the thing is that I got this equality with some examples. My question is:
1 - Is this true?
2 - If so, how can I prove this?
As I understood, by $\frac{\partial t_k(L(\textbf{x}))}{\partial l_i}$ you mean a formal partial derivative of the function $t_k(l_1(\textbf{x}),\dots,l_n(\textbf{x}))$ with respect to its $j$-th component $l_j$, that is a partial derivative of the function $t_k(\textbf{x})= t_k(x_1,\dots, x_n)$ with respect to its $j$-th component $x_j$, with each $x_i$ later substituted by $l_i(\textbf{x})$, that is with $\textbf{x}$ substituted by $L(\textbf{x}))$, which is the required equality
$$\frac{\partial t_k(L(\textbf{x}))}{\partial l_i}=\frac{\partial t_k(\textbf{x})}{\partial x_i}\Bigg|_{\textbf{x}=L(\textbf{x})}.$$