Answer
$\frac{(\ln 2)2^{\ln x}}{x}$
Work Step by Step
Let $y=2^{\ln x}$
Substitute $t= \ln x$
Then, $y= 2^{t}$
$\ln y= t \ln 2$
Differentiating both sides with respect to t, we get
$\frac{1}{y}\frac{dy}{dt}=\ln 2$
or $\frac{dy}{dt}=y \ln 2= 2^{t}\ln2= 2^{\ln x}\ln2$
$\frac{dt}{dx}=\frac{1}{x}$
According to the chain rule,
$\frac{dy}{dx}= \frac{dy}{dt}\cdot\frac{dt}{dx}=(\ln2 )2^{\ln x}\cdot\frac{1}{x}=\frac{(\ln 2)2^{\ln x}}{x}$
