#f(x)=1/sqrt(x)=1/x^(1/2)=x^(-1/2)#
Note that a square root is equivalent to raising an expression to the #1/2# power.
#f'(x)=lim_(h->0) (f(x+h)-f(x))/h#
#f(x+h)=1/sqrt(x+h)#
#f'(x)=lim_(h->0) (1/sqrt(x+h)-1/sqrt(x))/h#
Find the common denominator
#f'(x)=lim_(h->0) (1/sqrt(x+h)*sqrt(x)/sqrt(x)-1/sqrt(x)*sqrt(x+h)/sqrt(x+h))/h#
#f'(x)=lim_(h->0) (sqrt(x)/(sqrt(x)sqrt(x+h))-sqrt(x+h)/(sqrt(x)sqrt(x+h)))/h#
Consolidate the numerator of the complex fraction.
#f'(x)=lim_(h->0) ((sqrt(x)-sqrt(x+h))/(sqrt(x)sqrt(x+h)))/h#
Dividing fractions is equivalent to multiplying by the reciprocal
#f'(x)=lim_(h->0) 1/h*((sqrt(x)-sqrt(x+h))/(sqrt(x)sqrt(x+h)))#
Rationalize the numerator
#f'(x)=lim_(h->0) 1/h*((sqrt(x)-sqrt(x+h))/(sqrt(x)sqrt(x+h)))*(sqrt(x)+sqrt(x+h))/(sqrt(x)+sqrt(x+h)#
Simplify. Remember difference of perfect squares.
#f'(x)=lim_(h->0) 1/h*((x-(x+h))/(sqrt(x)sqrt(x+h)(sqrt(x)+sqrt(x+h))))#
Distribute the negative in the numerator
#f'(x)=lim_(h->0) 1/h*(x-x-h)/(sqrt(x)sqrt(x+h)(sqrt(x)+sqrt(x+h)))#
The #x's# resolve to zero.
#f'(x)=lim_(h->0) 1/h*(-h)/(sqrt(x)sqrt(x+h)(sqrt(x)+sqrt(x+h)))#
The #h's# can be cancelled.
#f'(x)=lim_(h->0) (-1)/(sqrt(x)sqrt(x+h)(sqrt(x)+sqrt(x+h)))#
Now we can substitute in 0 for #h#.
#(-1)/(sqrt(x)sqrt(x+0)(sqrt(x)+sqrt(x+0)))#
#(-1)/(sqrt(x)sqrt(x)(sqrt(x)+sqrt(x)))#
Manipulate the exponents
#(-1)/(x(2sqrt(x)))=(-1)/(x^1*2*x^(1/2))=(-1)/(x^(2/2)*2*x^(1/2))=(-1)/(2x^(3/2))#
