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

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

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

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

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

Solution steps

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

Treat

x, z

as constants

Apply the Sum/Difference Rule:

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

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

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

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

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

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

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

Simplify

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

\int_{0}^{1} cos^{2} (2 π x) d x

\frac{d}{d x} (x^{0.5} + 2 x^{- 0.5})

(2.6) (- 3.5)

4 sin (3 x)

\int \frac{2 - x}{4 x ^{2} + 4 x - 3} d x

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