What is expand (x+dx)^6 ?

The solution to expand (x+dx)^6 is x^6+6x^5dx+15x^4dx^2+20x^3dx^3+15x^2dx^4+6xdx^5+dx^6

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expand (x+dx)^6

expand

(x + d x)^{6}

x^{6} + 6 x^{5} d x + 15 x^{4} d x^{2} + 20 x^{3} d x^{3} + 15 x^{2} d x^{4} + 6 x d x^{5} + d x^{6}

Solution steps

(x + d x)^{6}

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 6 (i 6) x^{(6 - i)} d x^{i}

Expand summation

= \frac{6 !}{0 ! ( 6 - 0 ) !} x^{6} d x^{0} + \frac{6 !}{1 ! ( 6 - 1 ) !} x^{5} d x^{1} + \frac{6 !}{2 ! ( 6 - 2 ) !} x^{4} d x^{2} + \frac{6 !}{3 ! ( 6 - 3 ) !} x^{3} d x^{3} + \frac{6 !}{4 ! ( 6 - 4 ) !} x^{2} d x^{4} + \frac{6 !}{5 ! ( 6 - 5 ) !} x^{1} d x^{5} + \frac{6 !}{6 ! ( 6 - 6 ) !} x^{0} d x^{6}

Simplify

\frac{6 !}{0 ! ( 6 - 0 ) !} x^{6} d x^{0} : x^{6}

Simplify

\frac{6 !}{1 ! ( 6 - 1 ) !} x^{5} d x^{1} : 6 x^{5} d x

Simplify

\frac{6 !}{2 ! ( 6 - 2 ) !} x^{4} d x^{2} : 15 x^{4} d x^{2}

Simplify

\frac{6 !}{3 ! ( 6 - 3 ) !} x^{3} d x^{3} : 20 x^{3} d x^{3}

Simplify

\frac{6 !}{4 ! ( 6 - 4 ) !} x^{2} d x^{4} : 15 x^{2} d x^{4}

Simplify

\frac{6 !}{5 ! ( 6 - 5 ) !} x^{1} d x^{5} : 6 x d x^{5}

Simplify

\frac{6 !}{6 ! ( 6 - 6 ) !} x^{0} d x^{6} : d x^{6}

= x^{6} + 6 x^{5} d x + 15 x^{4} d x^{2} + 20 x^{3} d x^{3} + 15 x^{2} d x^{4} + 6 x d x^{5} + d x^{6}

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

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

\frac{56 y - 38}{3 ( y - 3 ) ( 5 y - 2 )}

\frac{s}{( s + 1 ) ( s + 2 )}

\frac{1}{5 ( s - 2 )}

What is expand (x+dx)^6 ?
The solution to expand (x+dx)^6 is x^6+6x^5dx+15x^4dx^2+20x^3dx^3+15x^2dx^4+6xdx^5+dx^6