begin{tikzpicture}
 begin{axis}[width=0.45textwidth]
  %small plot
 end{axis}
 begin{axis}[width=textwidth]
  %large plot
 end{axis}
end{tikzpicture}

documentclass{article}
usepackage{pgfplots}

begin{document}

begin{tikzpicture}
 begin{axis}[xshift=.5textwidth,yshift=2cm,width=0.45textwidth]
  %small plot
 end{axis}
 begin{axis}[width=textwidth]
  %large plot
 end{axis}
end{tikzpicture}

end{document}

documentclass{article}
usepackage{pgfplots}
usetikzlibrary{spy}
begin{document}

begin{tikzpicture}[spy using outlines={rectangle, magnification=3,connect spies}]
begin{axis}[grid=major,no markers,domain=-5:5,enlargelimits=false]
addplot {x^2};
addplot {x^3};

coordinate (spypoint) at (axis cs:0,0);
coordinate (spyviewer) at (axis cs:0.5,-90);
spy[width=6cm,height=1cm] on (spypoint) in node [fill=white] at (spyviewer);
end{axis}

end{tikzpicture}

end{document}

documentclass{standalone}
usepackage{pgfplots}
begin{document}
begin{tikzpicture}[remember picture]
    begin{axis}[width=textwidth]
    %large plot
    addplot {x^3};
    coordinate (insetPosition) at (rel axis cs:0.95,0.05);
    end{axis}
    begin{axis}[at={(insetPosition)},anchor={outer south east},footnotesize]
    %small plot
    addplot {2*x};
    end{axis}
end{tikzpicture}
end{document}

coordinate (insetPosition) at (axis cs:5:-140);