What is (\partial)/(\partial y)(y^2cos(xy^2)) ?

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

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(\partial)/(\partial y)(y^2cos(xy^2))

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

2 y cos (y^{2} x) - 2 y^{3} x sin (y^{2} x)

Solution steps

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

Treat

x

as a constant

Apply the Product Rule:

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

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

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

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

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

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

Simplify

2 y cos (x y^{2}) + (- sin (x y^{2}) \cdot 2 x y) y^{2} : 2 y cos (y^{2} x) - 2 y^{3} x sin (y^{2} x)

= 2 y cos (y^{2} x) - 2 y^{3} x sin (y^{2} x)

f (x) = x^{3} ln (x)

\int (sin (x) y) d y

\frac{d}{d x} (5 e^{- 3 x})

\int \frac{1}{4 e ^{- 5 x} + e ^{5 x}} d x

\int \frac{x}{y ^{2}} d y

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