Find function z(x,y)=11+3x^2y-2y^3x+5xy full differential.  Find function z(x,y)=11+3x^2y-2y^3x+5xy full differential.

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to remember that the total differential (full differential) of the function is the sum of prtial derivatives such that:

`dz = (del z)/(del x)dx + (del z)/(del y) dy`

You need to find the partial derivative  `(del z)/(del x)dx` , hence you need to differentiate the function with respect to x keeping y as constant such that:

`(del z)/(del x)dx = 6xy - 2y^3 + 5y`

You need to find the partial derivative `(del z)/(del y)dy` , hence you need to differentiate the function with respect to y keeping x as constant such that:

`(del z)/(del y)dy = 3x^2 - 6xy^2 + 5x`

You need to add the partial derivatives such that:

`dz = 6xy - 2y^3 + 5y + 3x^2 - 6xy^2 + 5x`

Hence, evaluating the full differential of function yields `dz = 6xy - 2y^3 + 5y + 3x^2 - 6xy^2 + 5x.`

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