What is expand (x+dx)^4 ?

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

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expand (x+dx)^4

expand

(x + d x)^{4}

x^{4} + 4 x^{3} d x + 6 x^{2} d x^{2} + 4 x d x^{3} + d x^{4}

Solution steps

(x + d x)^{4}

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 4 (i 4) x^{(4 - i)} d x^{i}

Expand summation

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

Simplify

\frac{4 !}{0 ! ( 4 - 0 ) !} x^{4} d x^{0} : x^{4}

Simplify

\frac{4 !}{1 ! ( 4 - 1 ) !} x^{3} d x^{1} : 4 x^{3} d x

Simplify

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

Simplify

\frac{4 !}{3 ! ( 4 - 3 ) !} x^{1} d x^{3} : 4 x d x^{3}

Simplify

\frac{4 !}{4 ! ( 4 - 4 ) !} x^{0} d x^{4} : d x^{4}

= x^{4} + 4 x^{3} d x + 6 x^{2} d x^{2} + 4 x d x^{3} + d x^{4}

(4 x^{2} y + 2 x) (3 ab c - d e f x)

\frac{81 ( n + 1 ) ^{2}}{4 n ^{2}}

\frac{1}{( s + 2 ) ( s ^{2} + 2 s + 3 )}

(1 + ln (x)) d x + (1 + ln (y)) d y

\frac{x + 8}{( x - 5 ) ( x + 5 )}

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