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

Answer 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/x^3/2