how to find the derivatives of this question?x/(square root of (3+square root of x-1)?

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beckden | High School Teacher | (Level 1) Educator

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`(d(x/(sqrt(3+sqrt(x-1)))))/(dx)`

First we will define `f(x)=sqrt(3+sqrt(x-1))`

And we can use the quotient property to take the derivative of `x/(f(x))`

`(d(x/(f(x))))/(dx)=((d(x))/(dx)f(x)-x((df(x))/(dx)))/(f(x))^2`

`(df(x))/(dx)=(d(x-1))/(dx)*(d(sqrt(x-1)))/(d(x-1))*(d(3+sqrt(x-1)))/(d(sqrt(x-1)))(d(sqrt(3+sqrt(x-1))))/(d(3+sqrt(x-1)))`

`(d(f(x)))/(dx)=(1)(1/(2sqrt(x-1)))(1)(1/(2(sqrt(3+sqrt(x-1)))))`

`(d(f(x)))/(dx)=1/(4(sqrt(x-1))(sqrt(3+sqrt(x-1))))`

So our final answer is

`(d(x))/(sqrt(3+sqrt(x-1)))=(sqrt(3+sqrt(x-1))-x(1/(4sqrt(x-1)sqrt(3+sqrt(x-1)))))/(sqrt(3+sqrt(x-1)))^2`

`=(sqrt(3+sqrt(x-1))-x/(4sqrt(x-1)sqrt(3+sqrt(x-1))))/(3+sqrt(x-1))`

`=(3x+12sqrt(x-1)-4)/(4(3+sqrt(x-1))^(3/2)sqrt(x-1))`

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