Find the derivative `(dy)/(du)` for `y = (u^2*e^u)/(1+ln u)`

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The function `y = (u^2*e^u)/(1 + ln u)`

Using the quotient rule and the product rule to find the derivative:

`y' = ((u^2*e^u)'(1 + ln u) -(u^2*e^u)(1 + ln u)')/(1 + ln u)^2`

=> `((2u*e^u + u^2*e^u)(1 + ln u) -(u^2*e^u)/u)/(1 + ln u)^2`

=> `((2u^2*e^u + u^3*e^u)(1 + ln u) -(u^2*e^u))/(u*(1 + ln u)^2)`

=> `((2u^2*e^u + u^3*e^u +2u^2*e^u*ln u + u^3*e^u*ln u - u^2*e^u))/(u*(1 + ln u)^2)`

=> `(u^2*e^u + u^3*e^u +2u^2*e^u*ln u + u^3*e^u*ln u)/(u*(1 + ln u)^2)`

The derivative `(dy)/(du) = (u^2*e^u + u^3*e^u +2u^2*e^u*ln u + u^3*e^u*ln u)/(u*(1 + ln u)^2)`

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