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:

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

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

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

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

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

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

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