Answer to Question #275331 in Calculus for Hafsa Tahsin

Question

Find partial z partial x and partial z partial y for

z ^ 4 - y ^ 2 + x ^ 2 = 6x ^ 2 * y ^ 3 * z ^ 7 .

Answer 1

$z ^ 4 - y ^ 2 + x ^ 2 = 6x ^ 2 y ^ 3 z ^ 7$

Taking partial derivate w.r.t $x$ both side for $z$ ($y$ will be treated as constant)

$4z^3\frac{\partial z}{\partial x}-0+2x=12xy^3z^7+42x^2y^3z^6 \frac{\partial z}{\partial x}\\ \Rightarrow \frac{\partial z}{\partial x}=\frac{12xy^3z^7}{4z^3-42x^2y^3z^6}$

Taking partial derivate w.r.t $y$ both side for $z$ ($x$ will be treated as constant)

$4z^3\frac{\partial z}{\partial y}-2y+0=18x^2y^2 z^7+42x^2y^3z^6 \frac{\partial z}{\partial y}\\ \Rightarrow \frac{\partial z}{\partial y}=\frac{12xy^3z^7}{4z^3-18x^2y^2 z^7}$