How do you find f'(x) using the limit definition given # sqrt(x−3)#?

1 Answer
Andrea S.
Jan 7, 2017

#f(x) = 1/(2sqrt(x-3))#

Explanation:

By definition the derivative of #f(x)# is:

#lim_(Deltax->0) (f(x+Deltax)-f(x))/(Deltax)#.

In our case:

#f'(x) = lim_(Deltax->0) (sqrt((x+Deltax-3))-sqrt(x-3))/(Deltax)#

Rationalize the numerator:

#f'(x) = lim_(Deltax->0) (sqrt((x+Deltax-3))-sqrt(x-3))/(Deltax)*(sqrt(x+Deltax-3)+sqrt(x-3))/(sqrt(x+Deltax-3)+sqrt(x-3)#

#f'(x) = lim_(Deltax->0) ((x+Deltax-3)-(x-3))/((Deltax)*(sqrt(x+Deltax-3)+sqrt(x-3))) = lim_(Deltax->0) 1/(sqrt(x+Deltax-3)+sqrt(x-3))=1/(2sqrt(x-3))#

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