What is expand (x(y+Δy)-y^3)/(Δy) ?

The solution to expand (x(y+Δy)-y^3)/(Δy) is x/Δ+x-(y^2)/Δ

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expand (x(y+Δy)-y^3)/(Δy)

expand

\frac{x ( y + Δ y ) - y ^{3}}{Δ y}

\frac{x}{Δ} + x - \frac{y ^{2}}{Δ}

Solution steps

\frac{x ( y + Δ y ) - y ^{3}}{Δ y}

Expand

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

= \frac{x y + Δ x y - y ^{3}}{Δ y}

Factor

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

= \frac{y ( x + Δ x - y ^{2} )}{Δ y}

Cancel the common factor:

y

= \frac{x + Δ x - y ^{2}}{Δ}

Apply the fraction rule:

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

\frac{x + Δ x - y ^{2}}{Δ} = \frac{x}{Δ} + \frac{Δ x}{Δ} - \frac{y ^{2}}{Δ}

= \frac{x}{Δ} + \frac{Δ x}{Δ} - \frac{y ^{2}}{Δ}

Cancel

\frac{Δ x}{Δ} : x

= \frac{x}{Δ} + x - \frac{y ^{2}}{Δ}

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

e^{a (x + 1)} + 2 x + 1

\frac{13 x + 2}{( 3 x + 1 ) ^{2}}

\frac{3 ( x + 1 )}{x - 1}

\frac{e ^{t} ( 1 - cos ( t ) )}{t}

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