expand (x(4ydx+2xdy)+y^3(3yx+5xdy))(x^αy^β)

Solution

expand

(x (4 y d x + 2 x d y) + y^{3} (3 y x + 5 x d y)) (x^{α} y^{β})

4 x^{1 + α} y^{1 + β} d x + 2 y^{β} x^{α + 2} d y + 3 x^{1 + α} y^{β + 4} + 5 x^{1 + α} y^{3 + β} d y

Solution steps

(x (4 y d x + 2 x d y) + y^{3} (3 y x + 5 x d y)) (x^{α} y^{β})

= (x (4 y d x + 2 x d y) + y^{3} (3 x y + 5 x d y)) (x^{α} y^{β})

Apply the distributive law:

a (b + c) = ab + a c

a = (x^{α} y^{β}), b = x (4 y d x + 2 x d y), c = y^{3} (3 y x + 5 x d y)

= (x^{α} y^{β}) x (4 y d x + 2 x d y) + (x^{α} y^{β}) y^{3} (3 y x + 5 x d y)

= x (x^{α} y^{β}) (4 y d x + 2 x d y) + y^{3} (x^{α} y^{β}) (3 x y + 5 x d y)

Simplify

x (x^{α} y^{β}) (4 y d x + 2 x d y) + y^{3} (x^{α} y^{β}) (3 x y + 5 x d y) : 4 x^{1 + α} y^{1 + β} d x + 2 y^{β} x^{α + 2} d y + 3 x^{1 + α} y^{β + 4} + 5 x^{1 + α} y^{3 + β} d y

= 4 x^{1 + α} y^{1 + β} d x + 2 y^{β} x^{α + 2} d y + 3 x^{1 + α} y^{β + 4} + 5 x^{1 + α} y^{3 + β} d y

e x p an d \frac{- 3 x ^{2} - 3 x - 10}{x ^{2} ( - 3 x ^{2} + 24 x )}

s im pl i f y 3 x (3 x^{2} - 8 3 x)

e x p an d (\frac{5 x}{6} - \frac{4 y}{3})^{2}

e x p an d (m + 5, n - 6) y (11, 3)

e x p an d \frac{( ( x + h ) ^{n} - x ^{n} )}{h}