(d^4)/(dx^4)((y)(e^x))

解

\frac{d ^{4}}{d x ^{4}} ((y) (e^{x}))

e^{x} y + 4 e^{x} \frac{d y}{d x} + 6 e^{x} \frac{d ^{2} y}{d x ^{2}} + 4 e^{x} \frac{d ^{3} y}{d x ^{3}} + e^{x} \frac{d ^{4} y}{d x ^{4}}

解答ステップ

\frac{d ^{4}}{d x ^{4}} ((y) (e^{x}))

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

= \frac{d ^{3}}{d x ^{3}} (\frac{d y}{d x} e^{x} + e^{x} y)

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

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

\frac{d}{d x} (e^{x} y + 2 e^{x} \frac{d y}{d x} + e^{x} \frac{d ^{2} y}{d x ^{2}}) = e^{x} y + 3 e^{x} \frac{d y}{d x} + 3 e^{x} \frac{d ^{2} y}{d x ^{2}} + e^{x} \frac{d ^{3} y}{d x ^{3}}

= \frac{d}{d x} (e^{x} y + 3 e^{x} \frac{d y}{d x} + 3 e^{x} \frac{d ^{2} y}{d x ^{2}} + e^{x} \frac{d ^{3} y}{d x ^{3}})

\frac{d}{d x} (e^{x} y + 3 e^{x} \frac{d y}{d x} + 3 e^{x} \frac{d ^{2} y}{d x ^{2}} + e^{x} \frac{d ^{3} y}{d x ^{3}}) = e^{x} y + 4 e^{x} \frac{d y}{d x} + 6 e^{x} \frac{d ^{2} y}{d x ^{2}} + 4 e^{x} \frac{d ^{3} y}{d x ^{3}} + e^{x} \frac{d ^{4} y}{d x ^{4}}

= e^{x} y + 4 e^{x} \frac{d y}{d x} + 6 e^{x} \frac{d ^{2} y}{d x ^{2}} + 4 e^{x} \frac{d ^{3} y}{d x ^{3}} + e^{x} \frac{d ^{4} y}{d x ^{4}}