Apparently I forgot how to do take a derivative.. This is the score of some Poisson distribution random variables: $\text{score}(x; \lambda) = \frac{x}{\lambda} - 1$.
Now I want to take the derivative according to the parameter $\lambda$. $\frac{\partial score(x; \lambda)}{\partial \lambda}$.
Remember the power rule, which states that $\frac{d}{dx} x^n=nx^{n-1}$. In your case, you are differentiating w.r.t lambda. Also remember that $\frac{d}{dx}cf(x)=c\frac{d}{dx}f(x)$ and $\frac{d}{dx}c=0$ for some number $c$. Finally, $\frac{d}{dx}(f(x)+g(x))=\frac{d}{dx}f(x)+\frac{d}{dx}g(x)$. This gives us the answer, which is $-\frac{x}{\lambda^2}$. Note that we are treating $x$ as a number since this is not the variable that we are differentiating with respect to.