calculus

find the limit of f'(x) = 1/(√x) using the limit definition of derivative as x approaches 0.

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  1. 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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