What is expand (v+Δv)^3 ?

The solution to expand (v+Δv)^3 is v^3+3Δv^3+3Δ^2v^3+Δ^3v^3

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expand (v+Δv)^3

expand

(v + Δ v)^{3}

v^{3} + 3Δ v^{3} + 3 Δ^{2} v^{3} + Δ^{3} v^{3}

Solution steps

(v + Δ v)^{3}

Apply Perfect Cube Formula:

(a + b)^{3} = a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}

a = v, b = Δ v

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

Simplify

v^{3} + 3 v^{2} Δ v + 3 v (Δ v)^{2} + (Δ v)^{3} : v^{3} + 3Δ v^{3} + 3 Δ^{2} v^{3} + Δ^{3} v^{3}

= v^{3} + 3Δ v^{3} + 3 Δ^{2} v^{3} + Δ^{3} v^{3}

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What is expand (v+Δv)^3 ?
The solution to expand (v+Δv)^3 is v^3+3Δv^3+3Δ^2v^3+Δ^3v^3