I've found the following expression. It looks really simple - so it's driving me crazy, that I don' get it:
$(e^{3x}).(2)$ is simplified as $2e^{2x}$.
Similarly, $(2x+7).(3e^{3x})$ is simplified as $(6x + 21)e^{2x}$
My problem isn't with where $6x$ and $21$ come from, because the multiplication by $3$ in $3e^{3x}$ seems obvious, what isn't obvious to me is why the $3$ in the superscript of $e^{3x}$ is changed to $e^{2x}$ in both cases. Surely, $ (2). e^{3x}$, just gives you 2 lots of $e^{3x}$, why should the exponent change to $2x$?
The expressions are used in an example of using the Quotient Rule in differentiation:
$$y = \frac{e^{3x}}{2x+7}$$
Set $u(x) = e^{3x}$ and $v(x) = 2x+7$
$$\frac{du}{dx} = 3e^{3x}$$ and $$ \frac{dv}{dx} = 2 $$
then
$$\frac{dy}{dx} = \frac{(2x+7).(3e^{3x})-(e^{3x}).(2)}{(2x+7)^2}$$
This is then simplified to this
$$\frac{(6x+21)e^{2x} - 2e^{2x}}{(2x+7)^2}$$
and then this
$$\frac{(6x+19)e^{2x}}{(2x+7)^2}$$
This simplification is completely invalid, and likely a typo. The derivative should be
$$\frac {(6x+19)e^{3x}} {(2x + 7)^2}$$
You can confirm with WolframAlpha if you want to be sure.