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

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

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

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

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

Solution steps

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

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

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

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

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

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

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

Simplify

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

y = x^{2} - 3, y = 1

\frac{d}{d x} (x^{3} - 12 x y + 8 y^{3})

\int (x - 1) (x + 2) d x

f (x) = 5 x^{\frac{4}{5}} - 2 x^{\frac{2}{7}}

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