I want to differentiate the Gamma function using product rule:
$$\Gamma(x+1)=x\Gamma(x)$$
$$\frac{d}{dx}\Gamma(x+1)=\frac{d}{dx}x\Gamma(x)$$
$$=x\left(\frac{d}{dx}\Gamma(x)\right)+\Gamma(x)$$
If $\frac{d}{dx}\Gamma(x)=F(x),$ then we have
$$F(x+1)=xF(x)+\Gamma(x)$$
I was wondering if there were any possible way to solve for $F(x)$ in terms of $x$ from here.
Another route:
$$\Gamma(x)\Gamma(x+1/2)=2^{1-2x}\sqrt{\pi}\Gamma(2x)$$
Differentiated both sides and applied product rule.
$$F(x)\Gamma(x+1/2)+F(x+1/2)\Gamma(x)=\sqrt{\pi}(-2\ln(2)2^{1-2x}\Gamma(2x)+2^{1-2x}F(2x))$$
Any way we could make use of these formulas to differentiate the Gamma function?