find the derivaties y= (u^2 * l^u) / (l + ln^u)

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mlehuzzah | Student, Graduate | (Level 1) Associate Educator

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y= (u^2 * l^u) / (l + ln^u)

If I understand you correctly, you have a 2-variable function:
your independent variables are `u`  and `l` 
and your dependent variable is `y`

And you are looking for the partial derivatives `(del y)/(del u)`, and `(del y)/(del l)`

(I'm not completely sure this is your question... it is a little odd to have `l`   as a variable, but you do mention plural derivatives. Also, ` ` ln^u doesn't make sense... perhaps it is either `l^u`  or `"ln"u` )

When you take a partial derivative, you pick one variable, and treat it as a variable, and every other variable you treat as a constant.

Assuming `y=(u^2 * l^u)/(l+"ln"u)`

Take `u` as a variable and `l` as a constant
You need the quotient and product rules:
`(del)/(del u)(u^2 * l^u)=(2u)(l^u)+(u^2)(l^u "ln" l)`
`(del y)/(del u)=(( (2u)(l^u)+(u^2)(l^u "ln" l))(l+"ln"u)-(u^2 * l^u)((1)/(u)))/((1+"ln"u)^2)`

Take `u` as a constant and `l` as the variable:
`(del)/(del l)(u^2 * l^u)=u^2 * (u l^(u-1))=u^3 l^(u-1) `
`(del y)/(del l) = ((u^3 l^(u-1))(l+"ln"u)-(u^2 * l^u)(1) )/((1+"ln"u)^2)`

Assuming `y=(u^2 * l^u)/(l+l^u)`
`(del)/(del u)(u^2 * l^u)=(2u)(l^u)+(u^2)(l^u "ln" l)`
`(del y)/(del u)=(((2u)(l^u)+(u^2)(l^u "ln" l))((l+l^u))-((u^2 * l^u))((l^u "ln" l)) )/((l+l^u)^2 )`

`(del)/(del l)(u^2 * l^u)=u^2 * ul^(u-1)=u^3 l^(u-1) `
`(del y)/(del l) = ((u^3 l^(u-1)) ((l+l^u)) -((u^2 * l^u)) (1+u l^(u-1)) )/((l+l^u)^2)`

Assuming `y=(u^2 * "ln" u)/(1+"ln"u)`
Then you just have a derivative with respect to u, and you don't have to worry about partial derivatives

`(d)/(du) (u^2 * "ln" u) = (2u)("ln"u)+(u^2)((1)/(u))=2u"ln"u+u`
`(dy)/(du)= ((2u"ln"u+u)(1+"ln"u)-(u^2 * "ln" u)((1)/(u)))/((1+"ln"u)^2)`