Given: `y=sqrt((x-1)/(x^4+1))`

Take the logarithm of both sides of the equation. Then rewrite the equation using the Laws of Logarithms.

`lny=lnsqrt((x-1)/(x^4+1))`

`lny=(1)/(2)[ln(x-1)-ln(x^4+1)]`

`lny=(1)/(2)ln(x-1)-(1)/(2)(x^4+1)`

Take the derivative of both sides of the equations.

`(1)/(y)dy/dx=(1)/(2(x-1))-(1)/(2(x^4+1))(4x^3)`

`dy/dx=y[(1)/(2(x-1))-((4x^3)/(2(x^4+1)))]`

Substitute in for y using the original equation. The derivative is:

`dy/dx=sqrt((x-1)/(x^4+1))[(1)/(2(x-1))-(2x^3)/(x^4+1)]`

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Given: `y=sqrt((x-1)/(x^4+1))`

Take the logarithm of both sides of the equation. Then rewrite the equation using the Laws of Logarithms.

`lny=lnsqrt((x-1)/(x^4+1))`

`lny=(1)/(2)[ln(x-1)-ln(x^4+1)]`

`lny=(1)/(2)ln(x-1)-(1)/(2)(x^4+1)`

Take the derivative of both sides of the equations.

`(1)/(y)dy/dx=(1)/(2(x-1))-(1)/(2(x^4+1))(4x^3)`

`dy/dx=y[(1)/(2(x-1))-((4x^3)/(2(x^4+1)))]`

Substitute in for y using the original equation. The derivative is:

`dy/dx=sqrt((x-1)/(x^4+1))[(1)/(2(x-1))-(2x^3)/(x^4+1)]`