What is (\partial)/(\partial y)((x)(y^4+3y-4x^3-5x)) ?

The answer to (\partial)/(\partial y)((x)(y^4+3y-4x^3-5x)) is x(4y^3+3)

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(\partial)/(\partial y)((x)(y^4+3y-4x^3-5x))

\frac{\partial}{\partial y} ((x) (y^{4} + 3 y - 4 x^{3} - 5 x))

x (4 y^{3} + 3)

Solution steps

\frac{\partial}{\partial y} ((x) (y^{4} + 3 y - 4 x^{3} - 5 x))

Treat

x

as a constant

Take the constant out:

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

= x \frac{\partial}{\partial y} (y^{4} + 3 y - 4 x^{3} - 5 x)

Apply the Sum/Difference Rule:

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

= x (\frac{\partial}{\partial y} (y^{4}) + \frac{\partial}{\partial y} (3 y) - \frac{\partial}{\partial y} (4 x^{3}) - \frac{\partial}{\partial y} (5 x))

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

\frac{\partial}{\partial y} (3 y) = 3

\frac{\partial}{\partial y} (4 x^{3}) = 0

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

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

Simplify

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

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What is (\partial)/(\partial y)((x)(y^4+3y-4x^3-5x)) ?
The answer to (\partial)/(\partial y)((x)(y^4+3y-4x^3-5x)) is x(4y^3+3)