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Let f(x)=(sqrt x)-2/(sqrt x)+2 and f'(4)=? would be?

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good2beme | Student, Undergraduate | Honors

Posted February 25, 2013 at 1:14 AM via web

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Let f(x)=(sqrt x)-2/(sqrt x)+2 and f'(4)=? would be?

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embizze | High School Teacher | (Level 1) Educator Emeritus

Posted February 25, 2013 at 2:54 AM (Answer #1)

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Find `f'(4)` if `f(x)=(sqrt(x)-2)/(sqrt(x)+2)` :

Write as `f(x)=(x^(1/2)-2)/(x^(1/2)+2)` then use the quotient rule: `(f/g)'=(gf'-fg')/g^2`

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

Then `f'(4)=(2/sqrt(4))/(sqrt(4)+2)^2=1/16`

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`f'(4)=1/16`

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