Does there exist a monotonically decreasing function that is its own derivative? I know that $f(x) = e^x$ is its own derivative. It is a monotonically increasing function. It seems intuitively plausible to me that there might be a monotonically decreasing function with the same property. Does one exist?
$f(x) = -ce^x, c &gt; 0$
This isn't a particularly exciting answer, but it is the correct one. All functions that are their own derivatives are of the form $f(x) = ce^x, c \in \mathbb{R}$, as explained in this question: Prove that $C\exp(x)$ is the only set of functions for which $f(x) = f&#39;(x)$
The requirement yields the differential equation dy - ydx = 0, which has the only solutions y =  kexp(x), where k is an arbitrary constant, and f(x) = 0 for all x in the domain. Now f(x) = 0 is decreasing and if k &lt; 0,  kexp(x) is a decreasing function. and these are the only collection of functions for which f’(x) = f(x) with f(x) decreasing.