Differential Calculus - Question

[studentxlol]

Homework Statement

I don't understand how this can be written like this:

$$\frac{1}{(2x+1)^{1/2}}-\frac{1}{(2x+1)^{3/2}}=\frac{2x+1-1}{(2x+1)^{3/2}}=\frac{2x}{(2x+1)^{3/2}}$$


What's the rule which makes this possible and explain please.

 

[HallsofIvy]

The "rule" is one that you probably learned in the third grade: you add or subtract fractions by getting a "common denominator".

The first fraction has denominator $(2x+1)^{1/2}$. The second fraction has denominator $(2x+ 1)^{3/2}= (2x+1)^{1+ 1/2}= (2x+1)(2x+1)^{1/2}$.

In other words, the common denominator is $(2x+1)^{3/2}$ and you get it by multiplying the numerator and denominator of the first fraction by 2x+1.