What is (\partial)/(\partial y)(x^4sin(xy^3)) ?

The answer to (\partial)/(\partial y)(x^4sin(xy^3)) is 3x^5y^2cos(xy^3)

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(\partial)/(\partial y)(x^4sin(xy^3))

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

3 x^{5} y^{2} cos (x y^{3})

Solution steps

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

Treat

x

as a constant

Take the constant out:

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

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

Apply the chain rule:

cos (x y^{3}) \frac{\partial}{\partial y} (x y^{3})

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

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

= x^{4} cos (x y^{3}) \cdot 3 x y^{2}

Simplify

x^{4} cos (x y^{3}) \cdot 3 x y^{2} : 3 x^{5} y^{2} cos (x y^{3})

= 3 x^{5} y^{2} cos (x y^{3})

What is (\partial)/(\partial y)(x^4sin(xy^3)) ?
The answer to (\partial)/(\partial y)(x^4sin(xy^3)) is 3x^5y^2cos(xy^3)