Per capita income growth rate is difference between income growth rate and population growth rate? We know: $PCI = \frac{Y(Income)}{P(population)}$ and that
$PCI$ growth rate is $\frac{\Delta PCI}{PCI}$.

How is this equal to the difference between income growth rate and population growth rate?

In other words how is $\left(\frac{∆\frac{Y}{P}}{\frac{Y}P}\right) = (∆Y/Y) - (∆P/P)$ ?
Knowing that:
$$\frac{\Delta PCI}{PCI}=
\left(\frac{∆\frac{Y}{P}}{\frac{Y}P}\right) $$

If we rearrange the terms on the right hand side:

$$\frac{\Delta PCI}{PCI}=\left(\frac{\frac{\Delta Y}{Y}}{\frac{\Delta P}{P}}\right)$$

In order to get your last equation we would have to log the variables, for logged growth rates. This makes things easier because we can think of the variables in terms of percentages.

Logging the variables we get
$$\ln\left(\frac{\Delta PCI}{PCI}\right)=\ln\left(\frac{\frac{\Delta Y}{Y}}{\frac{\Delta P}{P}}\right)$$

$$\ln\left(\frac{\Delta PCI}{PCI}\right)=\ln\left(\frac{\Delta Y}{Y}\right)-\ln\left(\frac{\Delta P}{P}\right)$$

in more formal notation used in macroeconomics:

$$\hat{PCI}=\hat{y}-\hat{p}$$

hope this helps.