\input amstex
$$\aligned
\varphi(x)-\varphi(x+1) &=\displaystyle\Biggl(x+\ln\Biggl(\frac{\Gamma(x)}{\sqrt{2\pi}x^{x-1/2}}\Biggr)\Biggr) -  \left((x+1)+\ln\left(\frac{\Gamma(x+1)}{\sqrt{2\pi}(x+1)^{(x+1)-1/2}}\right)\right) \\
&=\displaystyle\ln\left(\frac{\Gamma(x)}{\sqrt{2\pi}x^{x-1/2}}\right) - 1- \ln\left(\frac{x\cdot \Gamma(x)}{\sqrt{2\pi}(x+1)^{x+1/2}}\right)\\
&= \displaystyle\ln\left(\frac{\Gamma(x)\sqrt{2\pi}(x+1)^{x+1/2}}{x\cdot \Gamma(x)\sqrt{2\pi}x^{x-1/2}}\right) - 1 \\
&=  \displaystyle\ln\left(\frac{(x+1)^{x+1/2}}{x^{x+1/2}}\right) -1 = \left(x+\frac{1}{2}\right)\ln\left(1+\frac{1}{x}\right)-1
\endaligned$$
\bye