Let us find the partial derivatives of the function
z=x4y3+8x2yz=x^4y^3+8x^2yz=x4y3+8x2y :
∂z∂x=∂x4∂xy3+8∂x2∂xy=4x3y3+16xy\frac{\partial z}{\partial x}= \frac{\partial x^4}{\partial x}y^3+8\frac{\partial x^2}{\partial x}y=4x^3y^3+16xy∂x∂z=∂x∂x4y3+8∂x∂x2y=4x3y3+16xy
∂z∂y=x4∂y3∂y+8x2∂y∂y=3x4y2+8x2\frac{\partial z}{\partial y}= x^4\frac{\partial y^3}{\partial y}+8x^2\frac{\partial y}{\partial y}=3x^4y^2+8x^2∂y∂z=x4∂y∂y3+8x2∂y∂y=3x4y2+8x2
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