How do I use the limit definition of derivative to find #f'(x)# for #f(x)=sqrt(x+3)# ?

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Answer 1 · 2018-05-25T23:10:07

#d/dx (sqrt(x+3)) = 1/ (2sqrt(x+3)#

By definition:

#d/dx (sqrt(x+3)) = lim_(h->0) (sqrt(x+h+3)-sqrt(x+3))/h#

#d/dx (sqrt(x+3)) = lim_(h->0) ((sqrt(x+h+3)-sqrt(x+3))/h)( (sqrt(x+h+3)+sqrt(x+3))/ (sqrt(x+h+3)+sqrt(x+3)))#

#d/dx (sqrt(x+3)) = lim_(h->0) ((x+h+3)-(x+3))/(h (sqrt(x+h+3)+sqrt(x+3))#

#d/dx (sqrt(x+3)) = lim_(h->0) h/(h (sqrt(x+h+3)+sqrt(x+3))#

#d/dx (sqrt(x+3)) = lim_(h->0) 1/ (sqrt(x+h+3)+sqrt(x+3)#

#d/dx (sqrt(x+3)) = 1/ (2sqrt(x+3)#