# How do you differentiate  x/(e^(2x))?

May 31, 2016

$\frac{1 - 2 x}{e} ^ \left(2 x\right)$

#### Explanation:

differentiate using the $\textcolor{b l u e}{\text{quotient and chain rules}}$

f(x) = g(x).h(x) then f'(x)=(h(x).g'(x)-g(x).h'(x))/(h(x)^2

d/dx(f(g(x))=f'(g(x)).g'(x)
$\text{-------------------------------------------------------------}$
$g \left(x\right) = x \Rightarrow g ' \left(x\right) = 1$

$h \left(x\right) = {e}^{2 x} \Rightarrow h ' \left(x\right) = {e}^{2 x} .2 = 2 {e}^{2 x}$
$\text{------------------------------------------------------------}$
Substitute these values into f'(x) in the quotient rule

$f ' \left(x\right) = \frac{{e}^{2 x} .1 - x .2 {e}^{2 x}}{{e}^{2 x}} ^ 2 = \frac{{e}^{2 x} - 2 x {e}^{2 x}}{e} ^ \left(4 x\right)$

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