Derivative of $\frac{e^x}{x!}$

Question

I am having a bit of trouble putting all the differentiation rules together with the following problem:

$$ \frac{d}{da} \Bigg(\frac{a^x}{x!}e^{-a}\Bigg)$$

Where $x$ is a discrete variable and $a$ is a real number.

Thank you for your help.

Answer 1

If $n$ is a constant, then $${d\over du}u^n=nu^{n-1}$$ so if $x$ is a constant then $${d\over da}a^x=xa^{x-1}$$ Also, $${d\over du}e^{-u}=-e^{-u}$$ so $${d\over da}e^{-a}=-e^{-a}$$ Now that you have the derivatives of $a^x$ and $e^{-a}$ (with respect to $a$), it's just a matter of applying the product rule to work out $${d\over da}a^xe^{-a}$$ And as noted in the comments, $x!$ is just a constant so $${d\over da}\left({a^x\over x!}e^{-a}\right)={1\over x!}{d\over da}a^xe^{-a}$$ Does that do it for you?