I can't find (lny-lny0)/t meaning I  cannot understand what (lny-lny0)/t means. Of course we use this to show growth rate. But in the mathematical sense, I cannot know what it is. Is it diferrentiaion?
Generally ln is used to represent the natural logarithm.  I know of no notation system that uses it to represent differentiation.  

Natural logarithm:  

$$\frac{\ln{y} - \ln{y_0}}{t}$$

Differentiation would look more like 

$$\frac{dy}{dt}$$

or 

$$f'(t)$$

It is common to take logarithms of growth functions because they involve exponents and logarithms can simplify equations with exponents.  In particular, 

$$ \ln{(1+g)^t} = t\ln{(1+g)}$$

is a basic property of logarithms.  

A Maclaurin expansion is an alternative way to write the equation.  See Taylor series and Maclaurin's series for more information.  A Maclaurin expansion does in fact take advantage of differentiation.  

This logarithm is referred to as natural because its base arises from normal mathematical calculations rather than being arbitrary, much like $\pi$.