d/(dy)(xysqrt(1-(x^2)/4-(y^2)/9))

解

\frac{d}{d y} (x y 1 - \frac{x ^{2}}{4} - \frac{y ^{2}}{9})

\frac{x ( 36 - 9 x ^{2} - 8 y ^{2} )}{6 36 - 9 x ^{2} - 4 y ^{2}}

解答ステップ

\frac{d}{d y} (x y 1 - \frac{x ^{2}}{4} - \frac{y ^{2}}{9})

x

を定数として扱う

定数を除去:

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

= x \frac{d}{d y} (y 1 - \frac{x ^{2}}{4} - \frac{y ^{2}}{9})

積の法則を適用:

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

f = y, g = 1 - \frac{x ^{2}}{4} - \frac{y ^{2}}{9}

= x (\frac{d y}{d y} 1 - \frac{x ^{2}}{4} - \frac{y ^{2}}{9} + \frac{d}{d y} (1 - \frac{x ^{2}}{4} - \frac{y ^{2}}{9}) y)

\frac{d y}{d y} = 1

\frac{d}{d y} (1 - \frac{x ^{2}}{4} - \frac{y ^{2}}{9}) = - \frac{2 y}{3 - 4 y ^{2} + 36 - 9 x ^{2}}

= x (1 \cdot 1 - \frac{x ^{2}}{4} - \frac{y ^{2}}{9} + (- \frac{2 y}{3 - 4 y ^{2} + 36 - 9 x ^{2}}) y)

簡素化

x (1 \cdot 1 - \frac{x ^{2}}{4} - \frac{y ^{2}}{9} + (- \frac{2 y}{3 - 4 y ^{2} + 36 - 9 x ^{2}}) y) : \frac{x ( 36 - 9 x ^{2} - 8 y ^{2} )}{6 36 - 9 x ^{2} - 4 y ^{2}}

= \frac{x ( 36 - 9 x ^{2} - 8 y ^{2} )}{6 36 - 9 x ^{2} - 4 y ^{2}}