(\partial)/(\partial y)((8x)/(4x^2+5y^2))

解

\frac{\partial}{\partial y} (\frac{8 x}{4 x ^{2} + 5 y ^{2}})

- \frac{80 x y}{( 4 x ^{2} + 5 y ^{2} ) ^{2}}

解答ステップ

\frac{\partial}{\partial y} (\frac{8 x}{4 x ^{2} + 5 y ^{2}})

x

を定数として扱う

定数を除去:

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

= 8 x \frac{\partial}{\partial y} (\frac{1}{4 x ^{2} + 5 y ^{2}})

指数の規則を適用する:

\frac{1}{a} = a^{- 1}

= 8 x \frac{\partial}{\partial y} ((4 x^{2} + 5 y^{2})^{- 1})

連鎖法則を適用:

- \frac{1}{( 4 x ^{2} + 5 y ^{2} ) ^{2}} \frac{\partial}{\partial y} (4 x^{2} + 5 y^{2})

= - \frac{1}{( 4 x ^{2} + 5 y ^{2} ) ^{2}} \frac{\partial}{\partial y} (4 x^{2} + 5 y^{2})

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

= 8 x (- \frac{1}{( 4 x ^{2} + 5 y ^{2} ) ^{2}} \cdot 10 y)

簡素化

8 x (- \frac{1}{( 4 x ^{2} + 5 y ^{2} ) ^{2}} \cdot 10 y) : - \frac{80 x y}{( 4 x ^{2} + 5 y ^{2} ) ^{2}}

= - \frac{80 x y}{( 4 x ^{2} + 5 y ^{2} ) ^{2}}