(d^3)/(dx^3)((y)(x^{10}-sec(x)))

解

\frac{d ^{3}}{d x ^{3}} ((y) (x^{10} - sec (x)))

x^{10} \frac{d ^{3} y}{d x ^{3}} + 30 x^{9} \frac{d ^{2} y}{d x ^{2}} + 270 x^{8} \frac{d y}{d x} + 720 x^{7} y - sec (x) \frac{d ^{3} y}{d x ^{3}} - 3 sec (x) tan (x) \frac{d ^{2} y}{d x ^{2}} + 3 sec (x) \frac{d y}{d x} - 6 sec^{3} (x) \frac{d y}{d x} + sec (x) tan (x) y - 6 sec^{3} (x) tan (x) y

解答ステップ

\frac{d ^{3}}{d x ^{3}} ((y) (x^{10} - sec (x)))

\frac{d}{d x} (y (x^{10} - sec (x))) = \frac{d y}{d x} (x^{10} - sec (x)) + (10 x^{9} - sec (x) tan (x)) y

= \frac{d ^{2}}{d x ^{2}} (\frac{d y}{d x} (x^{10} - sec (x)) + (10 x^{9} - sec (x) tan (x)) y)

\frac{d}{d x} (\frac{d y}{d x} (x^{10} - sec (x)) + (10 x^{9} - sec (x) tan (x)) y) = \frac{d ^{2} y}{d x ^{2}} (x^{10} - sec (x)) + 2 \frac{d y}{d x} (10 x^{9} - sec (x) tan (x)) + (90 x^{8} + sec (x) - 2 sec^{3} (x)) y

= \frac{d}{d x} (\frac{d ^{2} y}{d x ^{2}} (x^{10} - sec (x)) + 2 \frac{d y}{d x} (10 x^{9} - sec (x) tan (x)) + (90 x^{8} + sec (x) - 2 sec^{3} (x)) y)

\frac{d}{d x} (\frac{d ^{2} y}{d x ^{2}} (x^{10} - sec (x)) + 2 \frac{d y}{d x} (10 x^{9} - sec (x) tan (x)) + (90 x^{8} + sec (x) - 2 sec^{3} (x)) y) = x^{10} \frac{d ^{3} y}{d x ^{3}} + 30 x^{9} \frac{d ^{2} y}{d x ^{2}} + 270 x^{8} \frac{d y}{d x} + 720 x^{7} y - sec (x) \frac{d ^{3} y}{d x ^{3}} - 3 sec (x) tan (x) \frac{d ^{2} y}{d x ^{2}} + 3 sec (x) \frac{d y}{d x} - 6 sec^{3} (x) \frac{d y}{d x} + sec (x) tan (x) y - 6 sec^{3} (x) tan (x) y

= x^{10} \frac{d ^{3} y}{d x ^{3}} + 30 x^{9} \frac{d ^{2} y}{d x ^{2}} + 270 x^{8} \frac{d y}{d x} + 720 x^{7} y - sec (x) \frac{d ^{3} y}{d x ^{3}} - 3 sec (x) tan (x) \frac{d ^{2} y}{d x ^{2}} + 3 sec (x) \frac{d y}{d x} - 6 sec^{3} (x) \frac{d y}{d x} + sec (x) tan (x) y - 6 sec^{3} (x) tan (x) y