(d^2)/(dx^2)(sqrt(r)+\sqrt[9]{r})

解

\frac{d ^{2}}{d x ^{2}} (r (x) + 9 r (x))

\frac{81 ( 2 \frac{d ^{2} r}{d x ^{2}} r - \frac{d r}{d x} ^{2} ) r ^{\frac{23}{18}} + 4 ( 9 \frac{d ^{2} r}{d x ^{2}} r ^{\frac{8}{9}} 9 r - 8 \frac{d r}{d x} ^{2} ) r ^{\frac{8}{9}}}{324 r ^{\frac{25}{9}}}

解答ステップ

\frac{d ^{2}}{d x ^{2}} (r + 9 r)

\frac{d}{d x} (r + 9 r) = \frac{\frac{d r}{d x}}{2 r} + \frac{\frac{d r}{d x}}{9 r ^{\frac{8}{9}}}

= \frac{d}{d x} (\frac{\frac{d r}{d x}}{2 r} + \frac{\frac{d r}{d x}}{9 r ^{\frac{8}{9}}})

\frac{d}{d x} (\frac{\frac{d r}{d x}}{2 r} + \frac{\frac{d r}{d x}}{9 r ^{\frac{8}{9}}}) = \frac{81 ( 2 \frac{d ^{2} r}{d x ^{2}} r - \frac{d r}{d x} ^{2} ) r ^{\frac{23}{18}} + 4 ( 9 \frac{d ^{2} r}{d x ^{2}} r ^{\frac{8}{9}} 9 r - 8 \frac{d r}{d x} ^{2} ) r ^{\frac{8}{9}}}{324 r ^{\frac{25}{9}}}

= \frac{81 ( 2 \frac{d ^{2} r}{d x ^{2}} r - \frac{d r}{d x} ^{2} ) r ^{\frac{23}{18}} + 4 ( 9 \frac{d ^{2} r}{d x ^{2}} r ^{\frac{8}{9}} 9 r - 8 \frac{d r}{d x} ^{2} ) r ^{\frac{8}{9}}}{324 r ^{\frac{25}{9}}}