1

In the following example, how can I shift the \Rightarrow slightly towards left so that the first terms of both equations are vertically aligned?

\documentclass[10pt]{article}

\usepackage{mathtools}

\usepackage{mathtools}
\usepackage{amsmath}
\usepackage{amsthm} %for proof 


\begin{document}

    \begin{align*}
    \frac{\partial V(u)}{\partial b}&=2.b.\sigma_{X}^{2}-2.\sigma_{XY}=0 \\ 
    \Rightarrow b &=\frac{\sigma_{XY}}{\sigma_{X}^{2}} = \beta^{*}
    \end{align*}




\end{document}
3

Since you are loading mathtools, you can use its \mathmakebox macro.

\documentclass[10pt]{article}

\usepackage{mathtools}

\usepackage{mathtools}
%\usepackage{amsmath}  %% already loaded by mathtools
\usepackage{amsthm} %for proof
\usepackage{calc}    %% provides \widthof


\begin{document}

    \begin{align*}
    \frac{\partial V(u)}{\partial b}&=2.b.\sigma_{X}^{2}-2.\sigma_{XY}=0 \\
    \mathmakebox[\widthof{$e\frac{\partial V(u)}{\partial b}$}][l]{\Rightarrow \hfill b} &=\frac{\sigma_{XY}}{\sigma_{X}^{2}} = \beta^{*}
    \end{align*}
\end{document}

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