Using the limit definition, how do you find the derivative of #f(x) = 2sqrtx#?

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Answer 1 · 2015-12-14T11:17:11

#f'(x) = 1/sqrt(x)#

#f'(x) = lim_(h->0) (f(x+h)-f(x))/h#

# = lim_(h->0)(2sqrt(x+h)-2sqrt(x))/h#

#= 2lim_(h->0)(sqrt(x+h)-sqrt(x))/h*(sqrt(x+h)+sqrt(x))/(sqrt(x+h)+sqrt(x))#

#=2lim_(h->0)(x+h-x)/(h(sqrt(x+h)+sqrt(x))#

#=2lim_(h->0) 1/(sqrt(x+h)+sqrt(x))#

#= 2*1/(sqrt(x+0)+sqrt(x))#

#=2*1/(2sqrt(x))#

#=1/sqrt(x)#