# What is the derivative of  e^x/2^x ?

Mar 18, 2017

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

#### Explanation:

Use the quotient rule $\left(\frac{u}{v}\right) ' = \frac{v \cdot u ' - u \cdot v '}{v} ^ 2$
Let $u = {e}^{x}$, $u ' = {e}^{x}$
Let $v = {2}^{x}$, $v ' = {2}^{x} \ln 2$

$\left(\frac{u}{v}\right) ' = \frac{{2}^{x} \cdot {e}^{x} - {e}^{x} \cdot {2}^{x} \ln 2}{{2}^{x}} ^ 2 = \frac{\left({2}^{x} {e}^{x}\right) \left(1 - \ln 2\right)}{{2}^{2 x}}$

Simplify using exponent rules ${x}^{m} / {x}^{2 m} = \frac{1}{x} ^ \left(2 m - m\right) = \frac{1}{x} ^ m$:

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

Mar 19, 2017

$\frac{d}{\mathrm{dx}} {e}^{x} / {2}^{x} = \left(1 - \ln 2\right) {e}^{x} / {2}^{x}$

#### Explanation:

Let $y = {e}^{x} / {2}^{x}$

Then take (Natural) logarithms of both sides; and use the rules of logs:

$\therefore \ln y = \ln \left\{{e}^{x} / {2}^{x}\right\}$
$\text{ } = \ln \left({e}^{x}\right) - \ln \left({2}^{x}\right)$
$\text{ } = x \ln \left(e\right) - x \ln \left(2\right)$
$\text{ } = x - x \ln \left(2\right)$
$\text{ } = x \left(1 - \ln \left(2\right)\right)$

Now, differentiate implicitly:

$\frac{1}{y} \frac{\mathrm{dy}}{\mathrm{dx}} \setminus = \left(1 - \ln 2\right)$
$\therefore \frac{\mathrm{dy}}{\mathrm{dx}} = \left(1 - \ln 2\right) y$
$\text{ } = \left(1 - \ln 2\right) {e}^{x} / {2}^{x}$

Mar 23, 2017

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

#### Explanation:

${e}^{x} / {2}^{x} = {\left(\frac{e}{2}\right)}^{x}$

The derivative of ${a}^{x}$ with respect to $x$ for any number $a$ is:

$\frac{d}{\mathrm{dx}} {a}^{x} = {a}^{x} \ln \left(a\right)$

Therefore:

$\frac{d}{\mathrm{dx}} {\left(\frac{e}{2}\right)}^{x} = {\left(\frac{e}{2}\right)}^{x} \ln \left(\frac{e}{2}\right)$

$\textcolor{w h i t e}{\text{XXXXX-}} = {\left(\frac{e}{2}\right)}^{x} \left(\ln e - \ln 2\right)$
$\textcolor{w h i t e}{\text{XXXXX-}} = {\left(\frac{e}{2}\right)}^{x} \left(1 - \ln 2\right)$