展開する (2(q+h)^4ln(q+h)-2q^4ln(q))/h

解

展開する

\frac{2 ( q + h ) ^{4} ln ( q + h ) - 2 q ^{4} ln ( q )}{h}

\frac{2 q ^{4} ln ( q + h )}{h} + 8 q^{3} ln (q + h) + 12 q^{2} h ln (q + h) + 8 q h^{2} ln (q + h) + 2 h^{3} ln (q + h) - \frac{2 q ^{4} ln ( q )}{h}

解答ステップ

\frac{2 ( q + h ) ^{4} ln ( q + h ) - 2 q ^{4} ln ( q )}{h}

(q + h)^{4} = q^{4} + 4 q^{3} h + 6 q^{2} h^{2} + 4 q h^{3} + h^{4}

= \frac{2 ln ( q + h ) ( q ^{4} + 4 q ^{3} h + 6 q ^{2} h ^{2} + 4 q h ^{3} + h ^{4} ) - 2 q ^{4} ln ( q )}{h}

拡張

2 (q^{4} + 4 q^{3} h + 6 q^{2} h^{2} + 4 q h^{3} + h^{4}) ln (q + h) - 2 q^{4} ln (q) : 2 q^{4} ln (q + h) + 8 q^{3} h ln (q + h) + 12 q^{2} h^{2} ln (q + h) + 8 q h^{3} ln (q + h) + 2 h^{4} ln (q + h) - 2 q^{4} ln (q)

= \frac{2 q ^{4} ln ( q + h ) + 8 q ^{3} h ln ( q + h ) + 12 q ^{2} h ^{2} ln ( q + h ) + 8 q h ^{3} ln ( q + h ) + 2 h ^{4} ln ( q + h ) - 2 q ^{4} ln ( q )}{h}

分数の規則を適用する:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{2 q ^{4} ln ( q + h ) + 8 q ^{3} h ln ( q + h ) + 12 q ^{2} h ^{2} ln ( q + h ) + 8 q h ^{3} ln ( q + h ) + 2 h ^{4} ln ( q + h ) - 2 q ^{4} ln ( q )}{h} = \frac{2 q ^{4} ln ( q + h )}{h} + \frac{8 q ^{3} h ln ( q + h )}{h} + \frac{12 q ^{2} h ^{2} ln ( q + h )}{h} + \frac{8 q h ^{3} ln ( q + h )}{h} + \frac{2 h ^{4} ln ( q + h )}{h} - \frac{2 q ^{4} ln ( q )}{h}

= \frac{2 q ^{4} ln ( q + h )}{h} + \frac{8 q ^{3} h ln ( q + h )}{h} + \frac{12 q ^{2} h ^{2} ln ( q + h )}{h} + \frac{8 q h ^{3} ln ( q + h )}{h} + \frac{2 h ^{4} ln ( q + h )}{h} - \frac{2 q ^{4} ln ( q )}{h}

\frac{8 q ^{3} h ln ( q + h )}{h} = 8 q^{3} ln (q + h)

\frac{12 q ^{2} h ^{2} ln ( q + h )}{h} = 12 q^{2} h ln (q + h)

\frac{8 q h ^{3} ln ( q + h )}{h} = 8 q h^{2} ln (q + h)

\frac{2 h ^{4} ln ( q + h )}{h} = 2 h^{3} ln (q + h)

= \frac{2 q ^{4} ln ( q + h )}{h} + 8 q^{3} ln (q + h) + 12 q^{2} h ln (q + h) + 8 q h^{2} ln (q + h) + 2 h^{3} ln (q + h) - \frac{2 q ^{4} ln ( q )}{h}