What is expand ((x+Δx)^2(x^2y+3y^3))/(Δx) ?

The solution to expand ((x+Δx)^2(x^2y+3y^3))/(Δx) is Δx^3y+3Δxy^3+2x^3y+6xy^3+(x^3y)/Δ+(3xy^3)/Δ

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expand ((x+Δx)^2(x^2y+3y^3))/(Δx)

expand

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

Δ x^{3} y + 3Δ x y^{3} + 2 x^{3} y + 6 x y^{3} + \frac{x ^{3} y}{Δ} + \frac{3 x y ^{3}}{Δ}

Solution steps

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

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

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

Factor

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

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

Cancel the common factor:

x

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

Expand

(Δ^{2} + 2Δ + 1) x y (x^{2} + 3 y^{2}) : Δ^{2} x^{3} y + 3 Δ^{2} x y^{3} + 2Δ x^{3} y + 6Δ x y^{3} + x^{3} y + 3 x y^{3}

= \frac{Δ ^{2} x ^{3} y + 3 Δ ^{2} x y ^{3} + 2Δ x ^{3} y + 6Δ x y ^{3} + x ^{3} y + 3 x y ^{3}}{Δ}

Apply the fraction rule:

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

\frac{Δ ^{2} x ^{3} y + 3 Δ ^{2} x y ^{3} + 2Δ x ^{3} y + 6Δ x y ^{3} + x ^{3} y + 3 x y ^{3}}{Δ} = \frac{Δ ^{2} x ^{3} y}{Δ} + \frac{3 Δ ^{2} x y ^{3}}{Δ} + \frac{2Δ x ^{3} y}{Δ} + \frac{6Δ x y ^{3}}{Δ} + \frac{x ^{3} y}{Δ} + \frac{3 x y ^{3}}{Δ}

= \frac{Δ ^{2} x ^{3} y}{Δ} + \frac{3 Δ ^{2} x y ^{3}}{Δ} + \frac{2Δ x ^{3} y}{Δ} + \frac{6Δ x y ^{3}}{Δ} + \frac{x ^{3} y}{Δ} + \frac{3 x y ^{3}}{Δ}

\frac{Δ ^{2} x ^{3} y}{Δ} = Δ x^{3} y

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

\frac{2Δ x ^{3} y}{Δ} = 2 x^{3} y

\frac{6Δ x y ^{3}}{Δ} = 6 x y^{3}

= Δ x^{3} y + 3Δ x y^{3} + 2 x^{3} y + 6 x y^{3} + \frac{x ^{3} y}{Δ} + \frac{3 x y ^{3}}{Δ}

(365, 8 - \frac{135000}{41} i) \cdot (0.6222 + 0.13334 i)

\frac{4 ( q - p )}{3 ( q + p )}

\frac{( 2 x + 3 ) ^{2}}{( 4 x - 1 ) ^{3}}

e^{- 2 - 5} (6 + 45)

\frac{1}{( s - 2 ) ^{2} ( s - 1 )}

What is expand ((x+Δx)^2(x^2y+3y^3))/(Δx) ?
The solution to expand ((x+Δx)^2(x^2y+3y^3))/(Δx) is Δx^3y+3Δxy^3+2x^3y+6xy^3+(x^3y)/Δ+(3xy^3)/Δ