What is (\partial)/(\partial x)((z)(x^4y+xz+yz^2)) ?

The answer to (\partial)/(\partial x)((z)(x^4y+xz+yz^2)) is z(z+4x^3y)

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(\partial)/(\partial x)((z)(x^4y+xz+yz^2))

\frac{\partial}{\partial x} ((z) (x^{4} y + x z + y z^{2}))

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

Solution steps

\frac{\partial}{\partial x} ((z) (x^{4} y + x z + y z^{2}))

Treat

z, y

as constants

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= z \frac{\partial}{\partial x} (x^{4} y + x z + y z^{2})

Apply the Sum/Difference Rule:

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

= z (\frac{\partial}{\partial x} (x^{4} y) + \frac{\partial}{\partial x} (x z) + \frac{\partial}{\partial x} (y z^{2}))

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

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

\frac{\partial}{\partial x} (y z^{2}) = 0

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

Simplify

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

2 (7 + 3)

(x - r \cdot cos (θ))^{2} + (y - r \cdot sin (θ))^{2}

y cos (x y) (y + x (\frac{d y}{d x}))

5 (3 - 6)

\frac{( x + 3 ) ^{2}}{( x + 5 ) ( x - 3 )}

What is (\partial)/(\partial x)((z)(x^4y+xz+yz^2)) ?
The answer to (\partial)/(\partial x)((z)(x^4y+xz+yz^2)) is z(z+4x^3y)