What is (\partial)/(\partial y)(x^5y^{10}cos(xy^2)) ?

The answer to (\partial)/(\partial y)(x^5y^{10}cos(xy^2)) is x^5(10y^9cos(xy^2)-2xy^{11}sin(xy^2))

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(\partial)/(\partial y)(x^5y^{10}cos(xy^2))

\frac{\partial}{\partial y} (x^{5} y^{10} cos (x y^{2}))

x^{5} (10 y^{9} cos (x y^{2}) - 2 x y^{11} sin (x y^{2}))

Solution steps

\frac{\partial}{\partial y} (x^{5} y^{10} cos (x y^{2}))

Treat

x

as a constant

Take the constant out:

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

= x^{5} \frac{\partial}{\partial y} (y^{10} cos (x y^{2}))

Apply the Product Rule:

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

f = y^{10}, g = cos (x y^{2})

= x^{5} (\frac{\partial}{\partial y} (y^{10}) cos (x y^{2}) + \frac{\partial}{\partial y} (cos (x y^{2})) y^{10})

\frac{\partial}{\partial y} (y^{10}) = 10 y^{9}

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

= x^{5} (10 y^{9} cos (x y^{2}) + (- sin (x y^{2}) \cdot 2 x y) y^{10})

Simplify

x^{5} (10 y^{9} cos (x y^{2}) + (- sin (x y^{2}) \cdot 2 x y) y^{10}) : x^{5} (10 y^{9} cos (x y^{2}) - 2 x y^{11} sin (x y^{2}))

= x^{5} (10 y^{9} cos (x y^{2}) - 2 x y^{11} sin (x y^{2}))

\frac{d}{d x} (25 e^{5 x})

d er i v a t i v e f (x) = sin^{3} (3 x)

\int 25 sin^{2} (x) d x

x \to 9 lim (1 - 4 x^{3})

d er i v a t i v e f (x) = arctan (sin (4 x))

What is (\partial)/(\partial y)(x^5y^{10}cos(xy^2)) ?
The answer to (\partial)/(\partial y)(x^5y^{10}cos(xy^2)) is x^5(10y^9cos(xy^2)-2xy^{11}sin(xy^2))