d/(dy)((2-y^2+x^2)e^{(1-x^2-(y^2)/4)})

解

\frac{d}{d y} ((2 - y^{2} + x^{2}) e^{(1 - x^{2} - \frac{y ^{2}}{4})})

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

解答ステップ

\frac{d}{d y} ((2 - y^{2} + x^{2}) e^{(1 - x^{2} - \frac{y ^{2}}{4})})

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 = 2 - y^{2} + x^{2}, g = e^{1 - x^{2} - \frac{y ^{2}}{4}}

= \frac{d}{d y} (2 - y^{2} + x^{2}) e^{1 - x^{2} - \frac{y ^{2}}{4}} + \frac{d}{d y} (e^{1 - x^{2} - \frac{y ^{2}}{4}}) (2 - y^{2} + x^{2})

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

\frac{d}{d y} (e^{1 - x^{2} - \frac{y ^{2}}{4}}) = - \frac{e ^{\frac{4 - 4 x ^{2} - y ^{2}}{4}} y}{2}

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

簡素化

(- 2 y) e^{1 - x^{2} - \frac{y ^{2}}{4}} + (- \frac{e ^{\frac{4 - 4 x ^{2} - y ^{2}}{4}} y}{2}) (2 - y^{2} + x^{2}) : - 2 e^{1 - x^{2} - \frac{y ^{2}}{4}} y - \frac{e ^{\frac{4 - 4 x ^{2} - y ^{2}}{4}} y ( 2 - y ^{2} + x ^{2} )}{2}

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