Answer
$-\dfrac{\ln 2}{ \sqrt{1+4^\theta}}$
Work Step by Step
We are given: $y=csch^{-1} 2^{\theta}$
Recall the formula: $\dfrac{d (csch^{-1} x)}{dx}=\dfrac{-1}{|x| \sqrt{1 + x^2}}$ and $\dfrac{d m^x}{dx}=m^x \ln m$
We need to use the product rule to get the differentiation:
Thus, $\dfrac{dy}{d \theta}=\dfrac{-1}{|2^{\theta}| \sqrt{1 + ((2)^{\theta})^2}}(2)^{\theta} \ln 2=-\dfrac{\ln 2}{ \sqrt{1+4^\theta}}$
