簡約 ((x+Δx)^3-x^3)/(Δx)

解

簡約

\frac{( x + Δ x ) ^{3} - x ^{3}}{Δ x}

Δ^{2} x^{2} + 3Δ x^{2} + 3 x^{2}

解答ステップ

\frac{( x + Δ x ) ^{3} - x ^{3}}{Δ x}

因数

(x + Δ x)^{3} - x^{3} : Δ x ((x + Δ x)^{2} + x (x + Δ x) + x^{2})

= \frac{Δ x ( ( x + Δ x ) ^{2} + x ( x + Δ x ) + x ^{2} )}{Δ x}

キャンセル

\frac{Δ x ( ( x + Δ x ) ^{2} + x ( x + Δ x ) + x ^{2} )}{Δ x} : (x + Δ x)^{2} + x (x + Δ x) + x^{2}

= (x + Δ x)^{2} + x (x + Δ x) + x^{2}

(x + Δ x)^{2} : x^{2} + 2Δ x^{2} + Δ^{2} x^{2}

= x^{2} + 2Δ x^{2} + Δ^{2} x^{2} + x (x + Δ x) + x^{2}

拡張

x (x + Δ x) : x^{2} + Δ x^{2}

= x^{2} + 2Δ x^{2} + Δ^{2} x^{2} + x^{2} + Δ x^{2} + x^{2}

簡素化

x^{2} + 2Δ x^{2} + Δ^{2} x^{2} + x^{2} + Δ x^{2} + x^{2} : Δ^{2} x^{2} + 3Δ x^{2} + 3 x^{2}

= Δ^{2} x^{2} + 3Δ x^{2} + 3 x^{2}