Limits and Derivatives

Limits and Derivatives

Question

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Find the derivative of f(x) = $space space space space space space space square root of x$

Given $space space space space space straight f left parenthesis straight x right parenthesis space equals space square root of straight x$

We have $space space space space space space space straight f space left parenthesis straight x right parenthesis space space space equals space limit as straight h rightwards arrow 0 of fraction numerator straight f left parenthesis straight x plus straight h right parenthesis minus straight f left parenthesis straight x right parenthesis over denominator straight h end fraction$

$italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic space italic equals italic space italic space stack l i m with h italic rightwards arrow italic 0 below fraction numerator square root of x italic plus h end root italic minus square root of x over denominator h end fraction italic equals stack l i m with h italic rightwards arrow italic 0 below fraction numerator square root of x italic plus h end root italic minus square root of x over denominator h end fraction x fraction numerator square root of x italic plus h end root italic plus square root of x over denominator square root of x italic plus h end root italic plus square root of x end fraction$

$space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space limit as straight h rightwards arrow 0 of fraction numerator straight x plus straight h minus straight x over denominator left parenthesis square root of straight x plus straight h end root plus square root of straight x end fraction equals limit as straight h rightwards arrow 0 of fraction numerator 1 over denominator square root of straight x plus straight h end root plus square root of straight x end fraction equals fraction numerator 1 over denominator square root of straight x plus square root of straight x end fraction equals fraction numerator 1 over denominator 2 square root of straight x end fraction$

Answer

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