What is expand (27e^x-3e^{3x})/((3+e^x)^4) ?

The solution to expand (27e^x-3e^{3x})/((3+e^x)^4) is -(3e^{2x}-9e^x)/(e^{3x)+9e^{2x}+27e^x+27}

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expand (27e^x-3e^{3x})/((3+e^x)^4)

expand

\frac{27 e ^{x} - 3 e ^{3 x}}{( 3 + e ^{x} ) ^{4}}

- \frac{3 e ^{2 x} - 9 e ^{x}}{e ^{3 x} + 9 e ^{2 x} + 27 e ^{x} + 27}

Solution steps

\frac{27 e ^{x} - 3 e ^{3 x}}{( 3 + e ^{x} ) ^{4}}

Factor

27 e^{x} - 3 e^{3 x} : - 3 e^{x} (e^{x} + 3) (e^{x} - 3)

= - \frac{3 e ^{x} ( e ^{x} + 3 ) ( e ^{x} - 3 )}{( 3 + e ^{x} ) ^{4}}

Cancel the common factor:

e^{x} + 3

= - \frac{3 e ^{x} ( e ^{x} - 3 )}{( e ^{x} + 3 ) ^{3}}

(e^{x} + 3)^{3} = e^{3 x} + 9 e^{2 x} + 27 e^{x} + 27

= - \frac{3 e ^{x} ( e ^{x} - 3 )}{27 e ^{x} + e ^{3 x} + 9 e ^{2 x} + 27}

Expand

3 e^{x} (e^{x} - 3) : 3 e^{2 x} - 9 e^{x}

= - \frac{- 9 e ^{x} + 3 e ^{2 x}}{27 e ^{x} + e ^{3 x} + 9 e ^{2 x} + 27}

(5 + 2 x^{\frac{1}{2}})^{4}

(x + 2 - 3) \cdot (x + 2 + 3)

2 (2 + 3)

100.2 + \frac{28.9}{x ( 1 + x )}

\frac{1}{( 5 x + 1 ) ^{3}}

What is expand (27e^x-3e^{3x})/((3+e^x)^4) ?
The solution to expand (27e^x-3e^{3x})/((3+e^x)^4) is -(3e^{2x}-9e^x)/(e^{3x)+9e^{2x}+27e^x+27}