Now I don't normaly do the quotient rule as it is just a version of the
product rule, but the quotient rule is:
d/dx [g(x)/h(x)] = [h(x)g'(x) - g(x)h'(x)]/[g(x)]^2
In your case g(x)=sqrt(x)-2, and h(x)=sqrt(x)+2, so:
g'(x)=(1/2)/sqrt(x), and h'(x)=(1/2)/sqrt(x).
Therefore applying the quotient rule:
d/dx [sqrt(x)-2)/(sqrt(x)+2)]
.........= [(sqrt(x)+2)/(2 sqrt(x)) - (sqrt(x)-2)/(2 sqrt(x))]/[sqrt(x)+2]^2
Which simplifies to:
d/dx [sqrt(x)-2)/(sqrt(x)+2)] = 2/(sqrt(x)[sqrt(x)+2]^2)
RonL