#### Answer

$(a)$. $$\frac{dy}{dx}=e^{-y}$$
$(b)$. The slope of the tangent line is $m=\frac{1}{2}$

#### Work Step by Step

$(a)$. Use implicit differentiation to find $\frac{dy}{dx}$ for $x=e^y$
Taking the derivative implicitly we get:
$$1=e^y\frac{dy}{dx}$$
Solve for $\frac{dy}{dx}$
$$\frac{dy}{dx}=\frac{1}{e^y}=e^{-y}$$
$(b)$. Finding slope of tangent line at the point $(2,\ln2)$ for the above function.
We do this by plugging in the point $(2,\ln2)$ into the derivative from part $(a)$. Hence:
$$\frac{dy}{dx}=e^{-(\ln2)}=e^{(\ln2^{-1})}=2^{-1}=\frac{1}{2}$$