In solving a Maximum Likelihood problem, I am trying to maximize the Poisson function in respect to its parameter x. Instead, I opt to maximize the logarithm of the function. The derivations have as follows. k = 0, 1, 2 ...
I am puzzled as k is non-negative.
Could you advise me where I err and how to proceed?
$\frac{d}{dx}(\log(e^{-x}))$ is $-1$, not $x$. If you want to do chain rule, the computation is $\frac{1}{e^{-x}} \cdot (-e^{-x}) = -1$.
However, it is much easier to to take logarithms before differentiating. $$\frac{d}{dx} \log \frac{x^k e^{-x}}{k!} = \frac{d}{dx}\left(k \log(x) - x - \log(k!)\right) = \frac{k}{x} - 1.$$
Setting this to zero gives $x=k$.