Determine-i-d-dx-x-1-x-ii-x-1-x-dx-

Question Number 2007 by Rasheed Soomro last updated on 29/Oct/15

D e t e r m i n e

(i) \frac{d}{d x} (x^{\frac{1}{x}}) (i i) \int x^{\frac{1}{x}} d x

Answered by Yozzi last updated on 29/Oct/15

L e t y = x^{1 / x} .

I f w e t a k e l o g a r i t h m s t o b a s e e, w h e r e

w e a s s u m e x > 0 (\Rightarrow y > 0), w e o b t a i n

l n y = {l n x}^{1 / x}

\Rightarrow l n y = \frac{1}{x} l n x (p o w e r r u l e o f l o g s)

D i f f e r e n t i a t i n g t h e e q u a t i o n i m p l i c i t l y

w . r . t x w e h a v e

\frac{1}{y} \times \frac{d y}{d x} = \frac{1}{x} \times \frac{1}{x} + \frac{- 1}{x^{2}} l n x

\Rightarrow \frac{d y}{d x} = y \frac{1}{x^{2}} (1 - l n x)

∴ \frac{d}{d x} (x^{1 / x}) = x^{1 / x} \times \frac{1}{x^{2}} (1 - l n x) = x^{\frac{1}{x} - 2} (1 - l n x)

(x > 0)

C o n t i n u e \dots (o n t o t h e i n t e g r a l w h i c h

l o o k s d i f f i c u l t t o f i n d)

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