find the derivative of the function $w(x)$, below. it may be to your advantage to simplify…

Explanation:

Step1: Rewrite the function

We know that $\frac{a^x}{b^x}=(\frac{a}{b})^x$. So $w(x) = 16(\frac{e}{2})^x$.

Step2: Apply the derivative formula

The derivative of $y = a\cdot b^x$ with respect to $x$ is $y'=a\cdot\ln(b)\cdot b^x$, where $a = 16$ and $b=\frac{e}{2}$. $w'(x)=16\cdot\ln(\frac{e}{2})\cdot(\frac{e}{2})^x$. Since $\ln(\frac{e}{2})=\ln(e)-\ln(2)=1 - \ln(2)$ and $(\frac{e}{2})^x=\frac{e^x}{2^x}$, we have $w'(x)=16(1 - \ln(2))\frac{e^x}{2^x}$.

Answer:

$16(1 - \ln(2))\frac{e^x}{2^x}$