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

解

\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}))

解答ステップ

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

x

を定数として扱う

定数を除去:

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

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

積の法則を適用:

(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})

簡素化

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}))