Question regarding differentiating with $e$ What is $\left[h(x) = \dfrac{3x-2}{e^x} \right]'?$
My textbook tackles this problem in this way:
$h'(x) = \dfrac{e^x\cdot3-(3x-2)e^x}{(e^x)^2} = \dfrac{3-(3x-2)}{e^x}$ etc...
I however don't understand how this is correct? If one of the $e^x$ in the numerator would have gotten cancelled I would have not problem,  but how come they're both cancelled?
 A: What you have above as $h'(x)$ is completely right. You surely know that for every real number $x$, $\text{e}^x\neq 0$. So if you factor it from the terms in the numerator and then cancel by one term in the denominator, you have a resulted fraction. In fact $$\frac{3\text{e}^x-(3x-2)\text{e}^x}{\text{e}^{2x}}=\frac{\text{e}^x(3-(3x-2))}{\text{e}^{x}\cdot\text{e}^{x}}=\frac{3-(3x-2)}{\text{e}^{x}}$$
A: \begin{align}
\text{INCORRECT}: & \qquad \frac{5a+5b}{5c} = \frac{a+5b}{c} \\[12pt]
\text{INCORRECT}: & \qquad \frac{5a+b}{5c} = \frac{a+b}{c} \\[12pt]
\text{CORRECT:} & \qquad \frac{5a+5b}{5c} = \frac{a+b}{c} 
\end{align}
You can cancel a factor that is common to the numerator and the denominator.  What was done in this last cancelation is this:
$$
\frac{5a+5b}{5c} = \frac{5(a+b)}{5c},
$$
followed by cancelation.  The number $5$ is a factor of the numerator since the numerator is $5$ times something.  The number $5$ is a factor of the denominator since the denominator is $5$ times something.
A: Recall the quotient rule that says that
$$
[f(g) / g(x)]' = \frac{f'(x)g(x) - f(x)g'(x)}{g(x)^2}
$$
Now with $g(x) = e^x$ also recall that $g'(x) = e^x$. And here $f(x) = 3x - 2$, so $f'(x) = 3$. Hence you get
$$
[f(g) / g(x)]' = \frac{3e^x - (3x-2)e^x}{(e^{x})^2} \stackrel{\star}{=}\frac{e^x[3 - (3x-2)]}{e^xe^x} = \frac{3 - (3x-2)}{e^x}
$$
In setp $(\star)$ you factor out an $e^x$ from the numerator.
A: I think what the OP has a problem with is this part of the problem:
$$\dfrac{e^x.3-e^x(3x-2)}{(e^x)^2} = \dfrac{e^x(5-3x)}{(e^x)(e^x)}=\dfrac{5-3x}{e^x}$$
That is - in the second step, we factorize the $e^x$ and thus yes, they "both" get cancelled.
