# Given xz^2+x^2y=5 and w=x^3y how determine derivative w.r.t. z as a function of x,y,z and detemine numerically when (x,y,z)=(1,1,2)?

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You need to find the partial derivative `(del w)/(del z), ` hence you need to differentiate `w= x^3y` with respect to z such that:

`(del w)/(del z) = (del (x^3y))/(del z) = (del w)/(del x)*(del x)/(del z)`

`(del w)/(del z) = (del (x^3y))/(del x)*(del x)/(del z)`

`(del w)/(del z) = 3x^2y*(del x)/(del z)`

You need to find `(del x)/(del z), ` hence you need to differentiate `xz^2+x^2y=5` with respect to z such that:

`2xz + (2xy+z^2)(del x)/(del z) = 0 =gt (2xy+z^2)(del x)/(del z) = -2xz`

`` `(del x)/(del z) = (-2xz)/(2xy+z^2)`

Hence, `(del w)/(del z) = 3x^2y*(-2xz)/(2xy+z^2)`

`(del w)/(del z) = (-6x^3yz)/(2xy+z^2)`

You need to evaluate `(del w)/(del z)` at `(x,y,z)=(1,1,2),` hence:

`(del w)/(del z)|_(1,1,2) = -12/(2 + 4)=gt(del w)/(del z)|_(1,1,2) = -12/6 =gt(del w)/(del z)|_(1,1,2) = -2`

**Hence, evaluating the function `(del w)/(del z) = (-6x^3yz)/(2xy+z^2)` at `(x,y,z)=(1,1,2)` yields `(del w)/(del z)|_(1,1,2) = -2.` **