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))
