What is (\partial)/(\partial y)(y^3cos(x+y)) ?

The answer to (\partial)/(\partial y)(y^3cos(x+y)) is 3y^2cos(x+y)-y^3sin(x+y)

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(\partial)/(\partial y)(y^3cos(x+y))

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

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

Solution steps

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

Treat

x

as a constant

Apply the Product Rule:

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

f = y^{3}, g = cos (x + y)

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

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

\frac{\partial}{\partial y} (cos (x + y)) = - sin (x + y)

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

Simplify

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

\int (2 x^{3} + 10 x^{2} - 3 x + 4) d x

y^{^{''}} - 6 y^{^{'}} = 4 t

\frac{d}{d x} (\frac{2 x + 1}{3 x - 1})

\int (x - 1) (x + 2) (x + 4) d x

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

What is (\partial)/(\partial y)(y^3cos(x+y)) ?
The answer to (\partial)/(\partial y)(y^3cos(x+y)) is 3y^2cos(x+y)-y^3sin(x+y)