What is (\partial)/(\partial z)(xyz+xy^4+yz^4+zx^4) ?

The answer to (\partial)/(\partial z)(xyz+xy^4+yz^4+zx^4) is x^4+xy+4yz^3

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(\partial)/(\partial z)(xyz+xy^4+yz^4+zx^4)

\frac{\partial}{\partial z} (x yz + x y^{4} + y z^{4} + z x^{4})

x^{4} + x y + 4 y z^{3}

Solution steps

\frac{\partial}{\partial z} (x yz + x y^{4} + y z^{4} + z x^{4})

Treat

x, y

as constants

Apply the Sum/Difference Rule:

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

= \frac{\partial}{\partial z} (x yz) + \frac{\partial}{\partial z} (x y^{4}) + \frac{\partial}{\partial z} (y z^{4}) + \frac{\partial}{\partial z} (z x^{4})

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

\frac{\partial}{\partial z} (x y^{4}) = 0

\frac{\partial}{\partial z} (y z^{4}) = 4 y z^{3}

\frac{\partial}{\partial z} (z x^{4}) = x^{4}

= x y + 0 + 4 y z^{3} + x^{4}

Simplify

= x^{4} + x y + 4 y z^{3}

\frac{\partial}{\partial z} (x sin (y))

\int_{0}^{2} sin (\frac{π t}{2}) d t

\int 18 t^{8} e^{- t^{9}} d t

\frac{\partial}{\partial z} (x y sin (\frac{π z}{3}))

\frac{\partial}{\partial x} (x e^{y} - 3 yz)

What is (\partial)/(\partial z)(xyz+xy^4+yz^4+zx^4) ?
The answer to (\partial)/(\partial z)(xyz+xy^4+yz^4+zx^4) is x^4+xy+4yz^3