Factorials and gamma function

I have some things to ask about factorials:

Is there a way to express the factorial function's derivative without using factorials?
Why is the slope of the gamma function at x=1 is equal to the Euler-Mascheroni Constant?
Does the factorial or gamma function have an analytic antiderivative?

#Calculus

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$