Is This Legal? 'e^x(-1) - (1 - x)e^x' to '(e^x)(-1)(-1)(x)(e

[Lime]

Is it correct to go from

e^x(-1) - (1 - x)e^x

to

(e^x) (-1) (-1) (x) (e^x)

:?:

 

[pka]

What does e^x(-1) mean?
\(\displaystyle \L
- e^x \quad ,\quad e^{(x - 1)} \quad ,\quad e^x - 1\quad \mbox{or what}\quad ??\)

 

[Lime]

Not sure. That's what it says in the book.

It's part of the second derivative. The first derivative is

f'(x) = (1 - x)/e^x

 

[pka]

\(\displaystyle \L
\begin{array}{rcl}
\frac{d}{{dx}}\left( {\frac{{1 - x}}{{e^x }}} \right) & = & \frac{{\left( { - 1} \right)e^x - (1 - x)e^x }}{{e^{2x} }} \\
& = & \frac{{ - e^x - e^x + xe^x }}{{e^{2x} }} \\
& = & \frac{{ - 2e^x + xe^x }}{{e^{2x} }} \\
& = & \frac{{ - 2 + x}}{{e^x }} \\
\end{array}\)

 

[Lime]

How do you go from (-1)e^x - (1 - x)e^x

to

-e^x - e^x + xe^x

:?:

 

[pka]

How do you go from (-1)e^x - (1 - x)e^x to
-e^x - e^x + xe^x

Very simple algebra:\(\displaystyle \L
\left( { - 1} \right)e^x - (1 - x)e^x = \left( { - e^x } \right) - (e^x - xe^x ).\)