derivative of sqrt((x^2-4(y^2-9)))

解

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

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

解答ステップ

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

連鎖法則を適用:

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

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

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

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

簡素化

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

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