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Question Number 82059 by jagoll last updated on 18/Feb/20

w h a t i s d e r i v a t i v e o f h = \sqrt{l n (x)}

b y f i r s t p r i n c i p l e m e t h o d

Answered by Henri Boucatchou last updated on 18/Feb/20

\frac{dh (x)}{dx} = \frac{d \sqrt{lnx}}{dx}

= \frac{\frac{dlnx}{dx}}{2 \sqrt{lnx}}

= \frac{1}{2 x \sqrt{lnx}}

Answered by mr W last updated on 18/Feb/20

h^{'} = \lim_{Δ x \to 0} \frac{\sqrt{\ln (x + Δ x)} - \sqrt{\ln x}}{Δ x}

= \lim_{Δ x \to 0} \frac{\sqrt{\ln x + \ln (1 + \frac{Δ x}{x})} - \sqrt{\ln x}}{Δ x}

= \sqrt{\ln x} \lim_{Δ x \to 0} \frac{\sqrt{1 + \frac{\ln (1 + \frac{Δ x}{x})}{\ln x}} - 1}{Δ x}

= \sqrt{\ln x} \lim_{Δ x \to 0} \frac{{1 + \frac{1}{2} [\frac{\ln (1 + \frac{Δ x}{x})}{\ln x}] + \dots} - 1}{Δ x}

= \sqrt{\ln x} \lim_{Δ x \to 0} {\frac{1}{2} [\frac{\ln (1 + \frac{Δ x}{x})}{Δ x \ln x}] + \dots}

= \sqrt{\ln x} \lim_{Δ x \to 0} {\frac{1}{2} [\frac{\ln {(1 + \frac{Δ x}{x})}^{\frac{x}{Δ x}}}{x \ln x}] + \dots}

= \sqrt{\ln x} \times \frac{1}{2} \times \frac{\ln e}{x \ln x}

= \frac{1}{2 x \sqrt{\ln x}}

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