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

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

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

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

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

Solution steps

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

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

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

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

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

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

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

Simplify

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

x \to - 2 lim (x + 5)

x \to 0 lim (\frac{e ^{x} - 1}{14 x})

\int y d y

f (x) = 5 x 5 - 4 x, at x = - 2

\frac{x}{x - 3}

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