Posted by John on Tuesday, October 19, 2010 at 8:30pm.
Given f(x)=1/√x
f'(x)
=Lim h->0 (f(x+h)-f(x))/h
=Lim h->0 (1/√(x+h)-1/√x))/h
subtract with common denominator
=Lim h->0 ((√x-√(x+h)/(h(√x √(x+h)))
multiply by conjugate of numerator, √(x)+√(x+h)
=Lim h->0 (x-(x+h))/(h(√x √(x+h))*(√x+&radic(x+h)))
subtract and cancel h
=Lim h->0 -1/(√x √(x+h)*(√x+&radic(x+h)))
Take limit h->0
=-1/x3/2
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