Question

find the derivative of the function
r = \frac{4e^{w}}{5w}
\frac{dr}{dw}=\frac{4we^{w}-4e^{w}}{5w^{2}}

Explanation:

Step1: Recall quotient - rule

The quotient - rule states that if $r=\frac{u}{v}$, then $\frac{dr}{dw}=\frac{u'v - uv'}{v^{2}}$, where $u = 4e^{w}$ and $v = 5w$.

Step2: Find $u'$ and $v'$

The derivative of $u = 4e^{w}$ with respect to $w$ is $u'=4e^{w}$ (since the derivative of $e^{w}$ is $e^{w}$ and by the constant - multiple rule). The derivative of $v = 5w$ with respect to $w$ is $v' = 5$.

Step3: Apply the quotient - rule

Substitute $u$, $u'$, $v$, and $v'$ into the quotient - rule formula:
\[

$$\begin{align*} \frac{dr}{dw}&=\frac{(4e^{w})\times(5w)-4e^{w}\times5}{(5w)^{2}}\\ &=\frac{20we^{w}-20e^{w}}{25w^{2}}\\ &=\frac{4we^{w}-4e^{w}}{5w^{2}} \end{align*}$$

\]

Answer:

$\frac{4we^{w}-4e^{w}}{5w^{2}}$