What is expand ((x+Δx)^3)/(Δx) ?

The solution to expand ((x+Δx)^3)/(Δx) is Δ^2x^2+3Δx^2+3x^2+(x^2)/Δ

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expand ((x+Δx)^3)/(Δx)

expand

\frac{( x + Δ x ) ^{3}}{Δ x}

Δ^{2} x^{2} + 3Δ x^{2} + 3 x^{2} + \frac{x ^{2}}{Δ}

Solution steps

\frac{( x + Δ x ) ^{3}}{Δ x}

(x + Δ x)^{3} = x^{3} + 3Δ x^{3} + 3 Δ^{2} x^{3} + Δ^{3} x^{3}

= \frac{x ^{3} + 3Δ x ^{3} + 3 Δ ^{2} x ^{3} + Δ ^{3} x ^{3}}{Δ x}

Factor out common term

x^{3}

= \frac{x ^{3} ( 1 + 3Δ + 3 Δ ^{2} + Δ ^{3} )}{Δ x}

Cancel the common factor:

x

= \frac{( Δ ^{3} + 3 Δ ^{2} + 3Δ + 1 ) x ^{2}}{Δ}

Expand

(Δ^{3} + 3 Δ^{2} + 3Δ + 1) x^{2} : Δ^{3} x^{2} + 3 Δ^{2} x^{2} + 3Δ x^{2} + x^{2}

= \frac{Δ ^{3} x ^{2} + 3 Δ ^{2} x ^{2} + 3Δ x ^{2} + x ^{2}}{Δ}

Apply the fraction rule:

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

\frac{Δ ^{3} x ^{2} + 3 Δ ^{2} x ^{2} + 3Δ x ^{2} + x ^{2}}{Δ} = \frac{Δ ^{3} x ^{2}}{Δ} + \frac{3 Δ ^{2} x ^{2}}{Δ} + \frac{3Δ x ^{2}}{Δ} + \frac{x ^{2}}{Δ}

= \frac{Δ ^{3} x ^{2}}{Δ} + \frac{3 Δ ^{2} x ^{2}}{Δ} + \frac{3Δ x ^{2}}{Δ} + \frac{x ^{2}}{Δ}

Cancel

\frac{Δ ^{3} x ^{2}}{Δ} : Δ^{2} x^{2}

= Δ^{2} x^{2} + \frac{3 Δ ^{2} x ^{2}}{Δ} + \frac{3Δ x ^{2}}{Δ} + \frac{x ^{2}}{Δ}

Cancel

\frac{3 Δ ^{2} x ^{2}}{Δ} : 3Δ x^{2}

= Δ^{2} x^{2} + 3Δ x^{2} + \frac{3Δ x ^{2}}{Δ} + \frac{x ^{2}}{Δ}

Cancel

\frac{3Δ x ^{2}}{Δ} : 3 x^{2}

= Δ^{2} x^{2} + 3Δ x^{2} + 3 x^{2} + \frac{x ^{2}}{Δ}

\frac{( - 2 x + 4 ) ^{7}}{( 3 x - 5 ) ^{5}}

x d y - 4 (y + x^{6}) d x

(π - 1) (π + 1)

(6 x + 5) (\frac{1}{2} e^{2 x})

(x^{2 + \frac{1}{x}})^{3}

What is expand ((x+Δx)^3)/(Δx) ?
The solution to expand ((x+Δx)^3)/(Δx) is Δ^2x^2+3Δx^2+3x^2+(x^2)/Δ