What is expand (x+dx)^5 ?

The solution to expand (x+dx)^5 is x^5+5x^4dx+10x^3dx^2+10x^2dx^3+5xdx^4+dx^5

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expand (x+dx)^5

expand

(x + d x)^{5}

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

Solution steps

(x + d x)^{5}

Apply binomial theorem:

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

a = x, b = d x

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

Expand summation

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

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

\frac{2}{3} (x + \frac{π}{2})

(5 a - 2 b + 3) + (4 b - 3 a + 5)

M (- 1, - 1)

(0.58) (6.79) 0

(x + 5) (x + Δ x)^{2} - 1

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