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Find `g'(4)` given that `f(4)=5`  and `f'(4)=1` and `g(x)=f(x)/x`.   

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user7230927 | (Level 1) Salutatorian

Posted June 29, 2013 at 1:32 AM via web

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Find `g'(4)` given that `f(4)=5`  and `f'(4)=1` and `g(x)=f(x)/x`. 

 

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mjripalda | High School Teacher | (Level 1) Senior Educator

Posted June 29, 2013 at 1:51 AM (Answer #1)

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`g(x)=(f(x))/x`

To solve for g'(4), take the derivative of g(x). To do so, apply quotient rule which is `(u/v)'=(v*u'-u*v')/v^2` .

`g'(x)=(x*(f(x))'-f(x)*x')/x^2`

Since the expression for the function f(x) is not given, then its derivative is expressed as f'(x) only.

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

`g'(x)=(x*f'(x)-f(x))/x^2`

Then, plug-in x=4 to get the value of g'(4).

`g'(4)=(4*f'(4)-f(4))/4^2`

`g'(4)=(4*f'(4)-f(4))/16`

Also, plug-in the given values of f(4) and f'(4) which are 5 and 1, respectively.

`g'(4)=(4*1-5)/16`

`g'(4)=(4-5)/16`

`g'(4)=(-1)/16`

`g'(4)=-1/16`

Hence, `g'(4)=-1/16` .

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