(\partial)/(\partial y)(e^{-y}(x^2-y^2))

解

\frac{\partial}{\partial y} (e^{- y} (x^{2} - y^{2}))

- e^{- y} (x^{2} - y^{2}) - 2 e^{- y} y

解答ステップ

\frac{\partial}{\partial y} (e^{- y} (x^{2} - y^{2}))

x

を定数として扱う

積の法則を適用:

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

f = e^{- y}, g = x^{2} - y^{2}

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

\frac{\partial}{\partial y} (e^{- y}) = - e^{- y}

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

= (- e^{- y}) (x^{2} - y^{2}) + (- 2 y) e^{- y}

簡素化

= - e^{- y} (x^{2} - y^{2}) - 2 e^{- y} y