What is (\partial)/(\partial y)(xsin(6x^2y)) ?

The answer to (\partial)/(\partial y)(xsin(6x^2y)) is 6x^3cos(6x^2y)

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(\partial)/(\partial y)(xsin(6x^2y))

\frac{\partial}{\partial y} (x sin (6 x^{2} y))

6 x^{3} cos (6 x^{2} y)

Solution steps

\frac{\partial}{\partial y} (x sin (6 x^{2} y))

Treat

x

as a constant

Take the constant out:

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

= x \frac{\partial}{\partial y} (sin (6 x^{2} y))

Apply the chain rule:

cos (6 x^{2} y) \frac{\partial}{\partial y} (6 x^{2} y)

= cos (6 x^{2} y) \frac{\partial}{\partial y} (6 x^{2} y)

\frac{\partial}{\partial y} (6 x^{2} y) = 6 x^{2}

= x cos (6 x^{2} y) \cdot 6 x^{2}

Simplify

x cos (6 x^{2} y) \cdot 6 x^{2} : 6 x^{3} cos (6 x^{2} y)

= 6 x^{3} cos (6 x^{2} y)

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What is (\partial)/(\partial y)(xsin(6x^2y)) ?
The answer to (\partial)/(\partial y)(xsin(6x^2y)) is 6x^3cos(6x^2y)