What is e^{-2x}(x/2 e^{2x}-1/4 e^{2x}) ?

The solution to e^{-2x}(x/2 e^{2x}-1/4 e^{2x}) is (2x-1)/4

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simplify e^{-2x}(x/2 e^{2x}-1/4 e^{2x})

simplify

e^{- 2 x} (\frac{x}{2} e^{2 x} - \frac{1}{4} e^{2 x})

\frac{2 x - 1}{4}

Solution steps

e^{- 2 x} (\frac{x}{2} e^{2 x} - \frac{1}{4} e^{2 x})

Apply exponent rule:

a^{- b} = \frac{1}{a ^{b}}

= \frac{1}{e ^{2 x}} (e^{2 x} \frac{x}{2} - \frac{1}{4} e^{2 x})

Multiply

\frac{x}{2} e^{2 x} : \frac{e ^{2 x} x}{2}

= \frac{1}{e ^{2 x}} (\frac{e ^{2 x} x}{2} - \frac{1}{4} e^{2 x})

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot ( \frac{x e ^{2 x}}{2} - \frac{1}{4} e ^{2 x} )}{e ^{2 x}}

1 \cdot (\frac{x e ^{2 x}}{2} - \frac{1}{4} e^{2 x}) = \frac{e ^{2 x} x}{2} - \frac{1}{4} e^{2 x}

= \frac{\frac{e ^{2 x} x}{2} - \frac{1}{4} e ^{2 x}}{e ^{2 x}}

Join

\frac{x e ^{2 x}}{2} - \frac{1}{4} e^{2 x} : \frac{2 e ^{2 x} x - e ^{2 x}}{4}

= \frac{\frac{2 e ^{2 x} x - e ^{2 x}}{4}}{e ^{2 x}}

Apply the fraction rule:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{x e ^{2 x} \cdot 2 - e ^{2 x}}{4 e ^{2 x}}

Factor out common term

e^{2 x}

= \frac{e ^{2 x} ( 2 x - 1 )}{4 e ^{2 x}}

Cancel the common factor:

e^{2 x}

= \frac{2 x - 1}{4}

13 e^{- 0.2 t}

600 e^{- 0.69}

1800 (1 + \frac{0.16}{2})^{- 4}

\frac{1}{2} (21 \cdot 1 0^{3} \cdot 0.1963 \cdot 10) (\frac{0.13}{10})^{2}

e^{- \frac{x ^{2}}{2}} \frac{d z}{d x}

What is e^{-2x}(x/2 e^{2x}-1/4 e^{2x}) ?
The solution to e^{-2x}(x/2 e^{2x}-1/4 e^{2x}) is (2x-1)/4