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

The answer to (\partial)/(\partial y)(4ysin(x^2y)) is 4(sin(yx^2)+yx^2cos(yx^2))

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(\partial)/(\partial y)(4ysin(x^2y))

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

4 (sin (y x^{2}) + y x^{2} cos (y x^{2}))

Solution steps

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

Treat

x

as a constant

Take the constant out:

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

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

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = y, g = sin (x^{2} y)

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

\frac{\partial}{\partial y} (y) = 1

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

= 4 (1 \cdot sin (x^{2} y) + cos (x^{2} y) x^{2} y)

Simplify

= 4 (sin (y x^{2}) + y x^{2} cos (y x^{2}))

\int_{2}^{9} \frac{2}{( 1 - x ) ^{\frac{2}{3}}} d x

\int_{0}^{5} 2 d x

\int (2 - x^{2})^{2} d x

\frac{d}{d x} (e^{x} + 2)

\frac{( x + 1 ) d y}{d x} (x + 1) y = 2 x e^{- x}

What is (\partial)/(\partial y)(4ysin(x^2y)) ?
The answer to (\partial)/(\partial y)(4ysin(x^2y)) is 4(sin(yx^2)+yx^2cos(yx^2))