What is expand (x+Δx)^5 ?

The solution to expand (x+Δx)^5 is x^5+5Δx^5+10Δ^2x^5+10Δ^3x^5+5Δ^4x^5+Δ^5x^5

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expand (x+Δx)^5

expand

(x + Δ x)^{5}

x^{5} + 5Δ x^{5} + 10 Δ^{2} x^{5} + 10 Δ^{3} x^{5} + 5 Δ^{4} x^{5} + Δ^{5} x^{5}

Solution steps

(x + Δ x)^{5}

Apply binomial theorem:

(a + b)^{n} = i = 0 \sum n (i n) a^{(n - i)} b^{i}

a = x, b = Δ x

= i = 0 \sum 5 (i 5) x^{(5 - i)} (Δ x)^{i}

Expand summation

= \frac{5 !}{0 ! ( 5 - 0 ) !} x^{5} (Δ x)^{0} + \frac{5 !}{1 ! ( 5 - 1 ) !} x^{4} (Δ x)^{1} + \frac{5 !}{2 ! ( 5 - 2 ) !} x^{3} (Δ x)^{2} + \frac{5 !}{3 ! ( 5 - 3 ) !} x^{2} (Δ x)^{3} + \frac{5 !}{4 ! ( 5 - 4 ) !} x^{1} (Δ x)^{4} + \frac{5 !}{5 ! ( 5 - 5 ) !} x^{0} (Δ x)^{5}

Simplify

\frac{5 !}{0 ! ( 5 - 0 ) !} x^{5} (Δ x)^{0} : x^{5}

Simplify

\frac{5 !}{1 ! ( 5 - 1 ) !} x^{4} (Δ x)^{1} : 5Δ x^{5}

Simplify

\frac{5 !}{2 ! ( 5 - 2 ) !} x^{3} (Δ x)^{2} : 10 Δ^{2} x^{5}

Simplify

\frac{5 !}{3 ! ( 5 - 3 ) !} x^{2} (Δ x)^{3} : 10 Δ^{3} x^{5}

Simplify

\frac{5 !}{4 ! ( 5 - 4 ) !} x^{1} (Δ x)^{4} : 5 Δ^{4} x^{5}

Simplify

\frac{5 !}{5 ! ( 5 - 5 ) !} x^{0} (Δ x)^{5} : Δ^{5} x^{5}

= x^{5} + 5Δ x^{5} + 10 Δ^{2} x^{5} + 10 Δ^{3} x^{5} + 5 Δ^{4} x^{5} + Δ^{5} x^{5}

(1 - cos (2 x))^{3}

(cot (x) - tan (x)) sin (2 x)

(x + 22)^{2}

\frac{3 x ^{2} - 5 x - 2}{x ( x - 1 ) ^{2}}

2 x y - 9 x^{2} + (2 y + x^{2} + 1) \frac{d y}{d x}

What is expand (x+Δx)^5 ?
The solution to expand (x+Δx)^5 is x^5+5Δx^5+10Δ^2x^5+10Δ^3x^5+5Δ^4x^5+Δ^5x^5