documentclass[10pt, twocolumn]{article}
usepackage{amsmath, amssymb}
begin{document}
begin{align*}
frac{ partial mathbb{E}[ X ] }{ partial j} approx  frac1{A_h} frac{partial}{partial j} left( frac23 A_K right)
    &= frac1{A_h} frac23 S_{ underline{ theta}_K }    \
& begin{aligned}[t]
    &approx frac1{A_h} frac23 S_{ underline{ theta}_h }approx frac1{A_h} frac23 underline{ hat{ theta} }_h \
&hspace{51pt} approx left( frac23 underline{ hat{ theta} }_h right)^{-1} frac23 underline{ hat{ theta} }_h = 1
end{aligned}
end{align*}
end{document}

documentclass{article}
usepackage{amsmath,amssymb,mathrsfs,bm}

newcommand{vect}[1]{bm{#1}} % change to underline{#1} if you prefer
newcommand{pder}[2]{frac{partial #1}{partial #2}}

begin{document}

begin{equation*}
begin{aligned}
pder{mathbb{E}[mathscr{X}]}{j}
=frac{1}{A_h}pder{}{j}biggl(frac{2}{3}A_Kbiggr)
&=frac{1}{A_h}frac{2}{3}S_{vect{theta}_K}
\
&approxbegin{aligned}[t]
frac{1}{A_h}frac{2}{3}S_{vect{theta}_h}
&approxfrac{1}{A_h}frac{2}{3}hat{vect{theta}}_h \
&approxbiggl(frac{2}{3}hat{vect{theta}}_hbiggr)^{!-1}
frac{2}{3}hat{vect{theta}}_h=1
end{aligned}
end{aligned}
end{equation*}

end{document}

documentclass{article}

usepackage{amsmath}

newcommand{pderiv}[2]{frac{partial #1}{partial #2}}

begin{document}

begin{alignat}{2}
  pderiv{E[X]}{j} approx
    frac{1}{A_h} pderiv{}{j} biggl( frac{2}{3} A_K biggr)
    &= frac{1}{A_h} frac{2}{3} S_{underline{theta}_K} \
    &approx frac{1}{A_h} frac{2}{3} S_{underline{theta}_K}
      &&approx frac{1}{A_h} frac{2}{3} underline{hat{theta}}_h \
    & &&approx biggl( frac{2}{3} underline{hat{theta}}_h biggr)^{-1}
          frac{2}{3} underline{hat{theta}}_h = 1
end{alignat}

end{document}

begin{align}
  pderiv{E[X]}{j} approx
    frac{1}{A_h} pderiv{}{j} biggl( frac{2}{3} A_K biggr)
    &= frac{1}{A_h} frac{2}{3} S_{underline{theta}_K} \
    &approx frac{1}{A_h} frac{2}{3} S_{underline{theta}_K}
      approx frac{1}{A_h} frac{2}{3} underline{hat{theta}}_h \
    & hphantom{{}approx frac{1}{A_h} frac{2}{3} S_{underline{theta}_K}}
      approx biggl( frac{2}{3} underline{hat{theta}}_h biggr)^{-1}
          frac{2}{3} underline{hat{theta}}_h = 1
end{align}

usepackage{mathtools}

% ...

begin{alignat}{2}
  pderiv{E[X]}{j} approx
    frac{1}{A_h} pderiv{}{j} biggl( frac{2}{3} A_K biggr)
    &= frac{1}{A_h} frac{2}{3} S_{underline{theta}_K}
      mathrlap{{}+ xyzklmno} \
    &approx frac{1}{A_h} frac{2}{3} S_{underline{theta}_K}
      &&approx frac{1}{A_h} frac{2}{3} underline{hat{theta}}_h \
    & &&approx biggl( frac{2}{3} underline{hat{theta}}_h biggr)^{-1}
          frac{2}{3} underline{hat{theta}}_h = 1
end{alignat}

documentclass[varwidth]{standalone}

begin{document}
[setlengtharraycolsep{1pt}
    begin{array}{rcl}
xxxx = yyyyy & = zzzz   &           \
             & = uuuu   & = vvvv    \
             &          & = wwww
    end{array}
]
end{document}

documentclass[10pt, twocolumn]{article}
usepackage{amsmath, amssymb}

begin{document}
[setlengtharraycolsep{1pt}
    begin{array}{rcl}
frac{partial mathbb{E}[X]}{partial j}
            approx  frac{1}{A_h} frac{partial}{partial j}
                left(frac{2}{3} A_K right)
    & = frac{1}{A_h} frac{2}{3} S_{underline{theta}_K}
        & approx frac{1}{A_h} frac{2}{3} S_{underline{theta}_h}    \
    &   & approx frac1{A_h} frac23 underline{ hat{ theta} }_h     \
    &   & approx left(frac{2}{3}
            underline{hat{theta}}_{h}right)^{-1}
                frac{2}{3}underline{hat{theta}}_{h} = 1
    end{array}
]
end{document}