How do you find f'(x) using the definition of a derivative for f(x)= x - sqrt(x) ?

1 Answer
George C.
Nov 1, 2015

Use limit definition of derivative to find:

f'(x) =1+1/(2sqrt(x))

Explanation:

f(x) = x - sqrt(x)

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

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

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

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

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

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

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

=1+1/(2sqrt(x))

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