What is expand (((dy)/(dx)+7x))^6 ?

The solution to expand (((dy)/(dx)+7x))^6 is

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expand (((dy)/(dx)+7x))^6

expand

((\frac{d y}{d x} + 7 x))^{6}

\frac{d y}{d x}^{6} + 42 x \frac{d y}{d x}^{5} + 735 x^{2} \frac{d y}{d x}^{4} + 6860 x^{3} \frac{d y}{d x}^{3} + 36015 x^{4} \frac{d y}{d x}^{2} + 100842 x^{5} \frac{d y}{d x} + 117649 x^{6}

Solution steps

(\frac{d y}{d x} + 7 x)^{6}

Apply binomial theorem:

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

a = \frac{d y}{d x}, b = 7 x

= i = 0 \sum 6 (i 6) \frac{d y}{d x}^{(6 - i)} (7 x)^{i}

Expand summation

= \frac{6 !}{0 ! ( 6 - 0 ) !} \frac{d y}{d x}^{6} (7 x)^{0} + \frac{6 !}{1 ! ( 6 - 1 ) !} \frac{d y}{d x}^{5} (7 x)^{1} + \frac{6 !}{2 ! ( 6 - 2 ) !} \frac{d y}{d x}^{4} (7 x)^{2} + \frac{6 !}{3 ! ( 6 - 3 ) !} \frac{d y}{d x}^{3} (7 x)^{3} + \frac{6 !}{4 ! ( 6 - 4 ) !} \frac{d y}{d x}^{2} (7 x)^{4} + \frac{6 !}{5 ! ( 6 - 5 ) !} \frac{d y}{d x}^{1} (7 x)^{5} + \frac{6 !}{6 ! ( 6 - 6 ) !} \frac{d y}{d x}^{0} (7 x)^{6}

Simplify

\frac{6 !}{0 ! ( 6 - 0 ) !} \frac{d y}{d x}^{6} (7 x)^{0} : \frac{d y}{d x}^{6}

Simplify

\frac{6 !}{1 ! ( 6 - 1 ) !} \frac{d y}{d x}^{5} (7 x)^{1} : 42 x \frac{d y}{d x}^{5}

Simplify

\frac{6 !}{2 ! ( 6 - 2 ) !} \frac{d y}{d x}^{4} (7 x)^{2} : 735 x^{2} \frac{d y}{d x}^{4}

Simplify

\frac{6 !}{3 ! ( 6 - 3 ) !} \frac{d y}{d x}^{3} (7 x)^{3} : 6860 x^{3} \frac{d y}{d x}^{3}

Simplify

\frac{6 !}{4 ! ( 6 - 4 ) !} \frac{d y}{d x}^{2} (7 x)^{4} : 36015 x^{4} \frac{d y}{d x}^{2}

Simplify

\frac{6 !}{5 ! ( 6 - 5 ) !} \frac{d y}{d x}^{1} (7 x)^{5} : 100842 x^{5} \frac{d y}{d x}

Simplify

\frac{6 !}{6 ! ( 6 - 6 ) !} \frac{d y}{d x}^{0} (7 x)^{6} : 117649 x^{6}

= \frac{d y}{d x}^{6} + 42 x \frac{d y}{d x}^{5} + 735 x^{2} \frac{d y}{d x}^{4} + 6860 x^{3} \frac{d y}{d x}^{3} + 36015 x^{4} \frac{d y}{d x}^{2} + 100842 x^{5} \frac{d y}{d x} + 117649 x^{6}

d (10)

\frac{109}{25 ( 10 x - 6 )}

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

\frac{27 ( x + 1 )}{x}

\frac{( d - c ) ( x - a )}{( b - a ) + c}

What is expand (((dy)/(dx)+7x))^6 ?
The solution to expand (((dy)/(dx)+7x))^6 is