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

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

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

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

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

Solution steps

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

Treat

z, x

as constants

Take the constant out:

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

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

Apply the Sum/Difference Rule:

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

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

\frac{\partial}{\partial y} (x^{2} y^{4} z) = 4 z x^{2} y^{3}

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

= z (4 z x^{2} y^{3} + 0)

Simplify

z (4 z x^{2} y^{3} + 0) : 4 z^{2} x^{2} y^{3}

= 4 z^{2} x^{2} y^{3}

(x e^{- x} + 1) e^{x}

\frac{8 ( 2 ln ( 2 ) - 1 )}{e ^{4}}

(2 x^{3} - 4 x + 5)^{6}

4 sin (2) (π - θ)

\frac{1 - n ^{2} x ^{2}}{( 1 + n ^{2} x ^{2} ) ^{2}}

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