x ln xy 1-find all partial derivatives  

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You should use the logarithmic identities such that:

`ln(xy) = ln x + ln y`

`f(x,y) = x*ln(xy) => f(x,y) = x*(ln x + ln y)`

`f(x,y) = x*ln x + x*ln y`

You need to find partial derivative `f_x` , hence, you need to differentiate the function with respect to x, keeping y constant, such that:

`f_x = ln x + x*1/x + lny`

`f_x = ln x + ln y+ 1 => f_x = 1 + ln(xy)`

You need to find partial derivative `f_y` , hence, you need to differentiate the function with respect to y, keeping x constant, such that:

`f_y = 0 + x*1/y => f_y = x/y`

Hence, evaluating the partial derivatives yields `f_x = 1 + ln(xy)`  and `f_y = x/y` .

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