What is (\partial)/(\partial x)(xyz+xy^3+yz^3+zx^3) ?

The answer to (\partial)/(\partial x)(xyz+xy^3+yz^3+zx^3) is y^3+yz+3x^2z

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(\partial)/(\partial x)(xyz+xy^3+yz^3+zx^3)

\frac{\partial}{\partial x} (x yz + x y^{3} + y z^{3} + z x^{3})

y^{3} + yz + 3 x^{2} z

Solution steps

\frac{\partial}{\partial x} (x yz + x y^{3} + y z^{3} + z x^{3})

Treat

y, z

as constants

Apply the Sum/Difference Rule:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{\partial}{\partial x} (x yz) + \frac{\partial}{\partial x} (x y^{3}) + \frac{\partial}{\partial x} (y z^{3}) + \frac{\partial}{\partial x} (z x^{3})

\frac{\partial}{\partial x} (x yz) = yz

\frac{\partial}{\partial x} (x y^{3}) = y^{3}

\frac{\partial}{\partial x} (y z^{3}) = 0

\frac{\partial}{\partial x} (z x^{3}) = 3 z x^{2}

= yz + y^{3} + 0 + 3 z x^{2}

Simplify

= y^{3} + yz + 3 x^{2} z

What is (\partial)/(\partial x)(xyz+xy^3+yz^3+zx^3) ?
The answer to (\partial)/(\partial x)(xyz+xy^3+yz^3+zx^3) is y^3+yz+3x^2z