展開する (1+x^2)(d^2)/(dx^2)((x+sqrt(x^2+1))^k)

解

展開する

(1 + x^{2}) \frac{d ^{2}}{d x ^{2}} ((x + x^{2} + 1)^{k})

\frac{d ^{2}}{d x ^{2}} ((x + x^{2} + 1)^{k}) + x^{2} \frac{d ^{2}}{d x ^{2}} ((x + x^{2} + 1)^{k})

解答ステップ

(1 + x^{2}) \frac{d ^{2}}{d x ^{2}} ((x + x^{2} + 1)^{k})

分配法則を適用する:

a (b + c) = ab + a c

a = \frac{d ^{2}}{d x ^{2}} ((x + x^{2} + 1)^{k}), b = 1, c = x^{2}

= \frac{d ^{2}}{d x ^{2}} ((x + x^{2} + 1)^{k}) 1 + \frac{d ^{2}}{d x ^{2}} ((x + x^{2} + 1)^{k}) x^{2}

= 1 \frac{d ^{2}}{d x ^{2}} ((x + x^{2} + 1)^{k}) + x^{2} \frac{d ^{2}}{d x ^{2}} ((x + x^{2} + 1)^{k})

乗算:

1 \frac{d ^{2}}{d x ^{2}} ((x + x^{2} + 1)^{k}) = \frac{d ^{2}}{d x ^{2}} ((x + x^{2} + 1)^{k})

= \frac{d ^{2}}{d x ^{2}} ((x + x^{2} + 1)^{k}) + x^{2} \frac{d ^{2}}{d x ^{2}} ((x + x^{2} + 1)^{k})