(\partial)/(\partial y)((x+y)^5+(x-y)^5)

解

\frac{\partial}{\partial y} ((x + y)^{5} + (x - y)^{5})

40 x^{3} y + 40 x y^{3}

解答ステップ

\frac{\partial}{\partial y} ((x + y)^{5} + (x - y)^{5})

簡素化

(x + y)^{5} + (x - y)^{5} : 2 x^{5} + 20 x^{3} y^{2} + 10 x y^{4}

= \frac{\partial}{\partial y} (2 x^{5} + 20 x^{3} y^{2} + 10 x y^{4})

x

を定数として扱う

和/差の法則を適用:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{\partial}{\partial y} (2 x^{5}) + \frac{\partial}{\partial y} (20 x^{3} y^{2}) + \frac{\partial}{\partial y} (10 x y^{4})

\frac{\partial}{\partial y} (2 x^{5}) = 0

\frac{\partial}{\partial y} (20 x^{3} y^{2}) = 40 x^{3} y

\frac{\partial}{\partial y} (10 x y^{4}) = 40 x y^{3}

= 0 + 40 x^{3} y + 40 x y^{3}

簡素化

= 40 x^{3} y + 40 x y^{3}