\
"We are given:"
\qquad \qquad \qquad \qquad \qquad \qquad \qquad g(x) \ = \ ( 1 / x^3 ) \cdot \sqrt{ 1 - e^{ 2x } }.
"We can rewrite" \ g(x) \ "to prepare it for differentiation:"
\qquad \qquad \qquad \qquad \qquad \qquad \qquad g(x) \ = \ x^{ -3 } \cdot ( 1 - e^{ 2x } )^{ 1/2 }.
"Using the Product Rule:"
\qquad \qquad g'(x) \ = \ x^{ -3 } \cdot [ ( 1 - e^{ 2x } )^{ 1/2 } ]' + [ x^{ -3 } ]' \cdot ( 1 - e^{ 2x } )^{ 1/2 }.
"Using the Chain Rule twice on the first differentiated quantity:"
g'(x) \ =
\ \ \ x^{ -3 } \cdot [ 1/2 ( 1 - e^{ 2x } )^{ -1/2 } ( 1 - e^{ 2x } )' ] + [ -3 x^{ -4 } ] \cdot ( 1 - e^{ 2x } )^{ 1/2 }
\qquad \qquad \ =
\ \ \ x^{ -3 } \cdot [ 1/2 ( 1 - e^{ 2x } )^{ -1/2 } ( 0 - 2 e^{ 2x } ) ] + ( -3 x^{ -4 } ) \cdot ( 1 - e^{ 2x } )^{ 1/2 }.
"Simplify:"
\qquad \qquad \ =
\ \ \ x^{ -3 } \cdot [ 1/2 ( 1 - e^{ 2x } )^{ -1/2 } ( - 2 e^{ 2x } ) ] + ( -3 x^{ -4 } ) \cdot ( 1 - e^{ 2x } )^{ 1/2 }
\qquad \qquad \ =
\ \ \ x^{ -3 } \cdot [ - ( 1 - e^{ 2x } )^{ -1/2 } ( e^{ 2x } ) ] + ( -3 x^{ -4 } ) \cdot ( 1 - e^{ 2x } )^{ 1/2 }.
"Continue simplification by pulling out lowest powers of same"
"quantities:"
\qquad \qquad \ =
\ \ \ - x^{ -4 } ( 1 - e^{ 2x } )^{ -1/2 } \cdot ( - x^1 ( e^{ 2x } ) + ( -3 ) \cdot ( 1 - e^{ 2x } )^{ 1 } )
\qquad \qquad \ = \ \ \ x^{ -4 } ( 1 - e^{ 2x } )^{ -1/2 } \cdot [ ( x e^{ 2x } ) + 3\cdot ( 1 - e^{ 2x } ) ]
\qquad \qquad \ = \ \ \ x^{ -4 } ( 1 - e^{ 2x } )^{ -1/2 } \cdot [ x e^{ 2x } + 3 - 3 e^{ 2x } ]
\qquad \qquad \ = \ \ \ x^{ -4 } ( 1 - e^{ 2x } )^{ -1/2 } \cdot [ 3 + x e^{ 2x } - 3 e^{ 2x } ]
\qquad \qquad \ = \ \ \ x^{ -4 } ( 1 - e^{ 2x } )^{ -1/2 } \cdot [ 3 + ( x- 3 ) e^{ 2x } ].
"Remove negative exponents, write one quantity as a square root:"
\qquad \qquad \ = \ \ \ [ 3 + ( x- 3 ) e^{ 2x } ] / { x^{ 4 } \sqrt{ 1 - e^{ 2x } } } \quad.
"This is our answer."
\
"Summarizing:"
\qquad \qquad \qquad \qquad \qquad \qquad \qquad g(x) \ = \ x^{ -3 } \cdot ( 1 - e^{ 2x } )^{ 1/2 }.
\qquad \qquad \qquad \qquad \qquad \qquad \qquad g'(x) \ = \ [ 3 + ( x- 3 ) e^{ 2x } ] / { x^{ 4 } \sqrt{ 1 - e^{ 2x } } } \quad.