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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