Question

If f(x)=√x for x>0, then, for x>a>0, f(x)-f(a)/x-a=

Answer 1

\[\frac{f(x)-f(a)}{x-a}=\frac{\sqrt x-\sqrt a}{x-a}\]
Notice that the denominator can be written as a difference of squares:
\[\frac{\sqrt x-\sqrt a}{x-a}=\frac{\sqrt x-\sqrt a}{\left(\sqrt x\right)^2-\left(\sqrt a\right)^2}=\frac{\sqrt x-\sqrt a}{\left(\sqrt x-\sqrt a\right)\left(\sqrt x+\sqrt a\right)}\]
Which reduces to
\[\frac{1}{\sqrt x+\sqrt a}\]