What is (\partial)/(\partial z)(xyz+xy^5+yz^5+zx^5) ?

The answer to (\partial)/(\partial z)(xyz+xy^5+yz^5+zx^5) is x^5+xy+5yz^4

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(\partial)/(\partial z)(xyz+xy^5+yz^5+zx^5)

\frac{\partial}{\partial z} (x yz + x y^{5} + y z^{5} + z x^{5})

x^{5} + x y + 5 y z^{4}

Solution steps

\frac{\partial}{\partial z} (x yz + x y^{5} + y z^{5} + z x^{5})

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^{5}) + \frac{\partial}{\partial z} (y z^{5}) + \frac{\partial}{\partial z} (z x^{5})

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

\frac{\partial}{\partial z} (x y^{5}) = 0

\frac{\partial}{\partial z} (y z^{5}) = 5 y z^{4}

\frac{\partial}{\partial z} (z x^{5}) = x^{5}

= x y + 0 + 5 y z^{4} + x^{5}

Simplify

= x^{5} + x y + 5 y z^{4}

f (x) = 2 e^{3 x}

- 4 e^{4 t}

\frac{\partial}{\partial y} (cos (x))

7 y^{^{''}} + 26 y = 0

\frac{- 5}{6 x ^{6}}

What is (\partial)/(\partial z)(xyz+xy^5+yz^5+zx^5) ?
The answer to (\partial)/(\partial z)(xyz+xy^5+yz^5+zx^5) is x^5+xy+5yz^4