Find the derivatives using logarithmic differentiation y=5 cube root x-1/x+1

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to find derivative of the given function `y = 5 root(3)((x-1)/(x+1)) ` using chain rule and quotient rule, hence, you do not need to use logarithmic differentiation in this situation since there is no exponent that contains variable x.

You need to convert the cube root into a power such that:

`y = 5*((x-1)/(x+1))^(1/3)`

`(dy)/(dx) = 5*(1/3)*((x-1)/(x+1))^(1/3 - 1)*((x-1)/(x+1))'`

`(dy)/(dx) = 5*(1/3)*((x-1)/(x+1))^(-2/3)*((x-1)'(x+1)-(x-1)(x+1)')/((x+1)^2)`

`(dy)/(dx) = 5*(1/3)*((x-1)/(x+1))^(-2/3)*(x+1-x+1)/((x+1)^2)`

`(dy)/(dx) = 10/(3((x-1)/(x+1))^(2/3)(x+1)^2)`

`(dy)/(dx) = 10/(3root(3)(((x-1)/(x+1))^2)(x+1)^2)`

Hence, differentiating the given function with respect to x yields `(dy)/(dx) = 10/(3root(3)(((x-1)/(x+1))^2)(x+1)^2).`