What is expand (y^3-v^3y^3)(ydv+vdy) ?

The solution to expand (y^3-v^3y^3)(ydv+vdy) is y^4dv+vy^3dy-v^3y^4dv-v^4y^3dy

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expand (y^3-v^3y^3)(ydv+vdy)

expand

(y^{3} - v^{3} y^{3}) (y d v + v d y)

y^{4} d v + v y^{3} d y - v^{3} y^{4} d v - v^{4} y^{3} d y

Solution steps

(y^{3} - v^{3} y^{3}) (y d v + v d y)

Apply FOIL method:

(a + b) (c + d) = a c + a d + b c + b d

a = y^{3}, b = - v^{3} y^{3}, c = y d v, d = v d y

= y^{3} y d v + y^{3} v d y + (- v^{3} y^{3}) y d v + (- v^{3} y^{3}) v d y

Apply minus-plus rules

+ (- a) = - a

= y^{3} y d v + v y^{3} d y - v^{3} y^{3} y d v - v^{3} v y^{3} d y

Simplify

y^{3} y d v + v y^{3} d y - v^{3} y^{3} y d v - v^{3} v y^{3} d y : y^{4} d v + v y^{3} d y - v^{3} y^{4} d v - v^{4} y^{3} d y

= y^{4} d v + v y^{3} d y - v^{3} y^{4} d v - v^{4} y^{3} d y

32 (x - (\frac{π}{4}))

π (- 42 + 2)

\frac{( 2 x - 1 ) ( 4 - 3 x )}{x}

ln (2) (x + y)

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

What is expand (y^3-v^3y^3)(ydv+vdy) ?
The solution to expand (y^3-v^3y^3)(ydv+vdy) is y^4dv+vy^3dy-v^3y^4dv-v^4y^3dy