How do you use the limit definition of the derivative to find the derivative of $f(x)=sqrtx$ ?

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Answer 1 · 2016-07-23T00:41:19

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

$f'(x) = lim_(Deltaxrarr0) (f(x+Deltax) - f(x))/(Deltax)$

$f'(x) = lim_(Deltaxrarr0) (sqrt(x+Deltax) - sqrt(x))/(Deltax)$

$f'(x) = lim_(Deltaxrarr0) (sqrt(x+Deltax) - sqrt(x))/(Deltax) * (sqrt(x+Deltax) + sqrt(x))/(sqrt(x+Deltax) + sqrt(x))$

$f'(x) = lim_(Deltaxrarr0) (cancel(Deltax))/(cancel(Deltax)(sqrt(x+Deltax) + sqrt(x))$

$f'(x) = lim_(Deltaxrarr0) 1/(sqrt(x+Deltax) + sqrt(x)) = 1/(2sqrt(x))$

How do you use the limit definition of the derivative to find the derivative of f(x)=sqrtxf(x)=sqrtx?