Question

Help !!!
See question in image.
Please show workings .
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Answer 1

see explanation

Explanation:

Given f(x) then the derivative f'(x) is

f'(x) = lim( h tends to 0 )
`(f(x+h)-f(x))/(h)`

= lim( h to 0 )
`(x+h+(1)/(x+h )-(x+(1)/(x) ) )/(h)`

= lim(h to 0)
`(x+h+(1)/(x+h)-x-(1)/(x) )/(h)`

= (lim(h to 0)
`(h+(1)/(x+h)-(1)/(x) )/(h)`

= lim( h to 0 )
`(hx(x+h)+x-(x+h))/(hx(x+h))`

= lim( h to 0 )
`(hx(x+h)+x-x-h)/(hx(x+h))`

= lim(h to 0 )
`(hx(x+h))/(hx(x+h))` -
`(h)/(hx(x+h))`

Cancel the numerator/denominator of first fraction and h in the second

= lim ( h to 0 ) 1 -
`(1)/(x(x+h))` ( let h go to zero ), then

f'(x) = 1 -
`(1)/(x^2)`