The limit definition of the derivative of a function #f(x)# is:
#f'(x) = lim_(Deltax->0) (f(x+Deltax)-f(x))/(Deltax) = lim_(Deltax->0) (Deltaf)/(Deltax)#
Let's calculate the function variation between #x# and #x+Deltax#:
#Delta f = 2/sqrt(x+Deltax) -2/sqrt(x)= 2(1/sqrt(x+Deltax) -1/sqrt(x))= 2(sqrt(x)- sqrt(x+Deltax))/(sqrt(x)sqrt(x+Deltax))#
We can now rationalize the numerator:
#Delta f = 2(sqrt(x)- sqrt(x+Deltax))/(sqrt(x)sqrt(x+Deltax)) * (sqrt(x)+ sqrt(x+Deltax))/(sqrt(x)+ sqrt(x+Deltax))= 2(x-(x+Deltax))/(xsqrt(x+Deltax) + sqrt(x) (x+Deltax))= (-2Deltax)/(xsqrt(x+Deltax) + sqrt(x) (x+Deltax))#
Divide by #Deltax#:
#(Deltaf)/(Deltax) = (-2)/(xsqrt(x+Deltax) + sqrt(x) (x+Deltax))#
and passing to the limit for #Deltax->0#:
#lim_(Deltax->0) (-2)/(xsqrt(x+Deltax) + sqrt(x) (x+Deltax)) = (-2)/(xsqrt(x) +sqrt(x)x) = (-2)/(2xsqrt(x)) = -1/(xsqrt(x))#
