What is expand (vx-2x)(vdx+xdv) ?

The solution to expand (vx-2x)(vdx+xdv) is v^2xdx+vx^2dv-2vxdx-2x^2dv

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expand (vx-2x)(vdx+xdv)

expand

(vx - 2 x) (v d x + x d v)

v^{2} x d x + v x^{2} d v - 2 vx d x - 2 x^{2} d v

Solution steps

(vx - 2 x) (v d x + x d v)

Apply FOIL method:

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

a = vx, b = - 2 x, c = v d x, d = x d v

= vxv d x + vxx d v + (- 2 x) v d x + (- 2 x) x d v

Apply minus-plus rules

+ (- a) = - a

= vvx d x + vxx d v - 2 vx d x - 2 xx d v

Simplify

vvx d x + vxx d v - 2 vx d x - 2 xx d v : v^{2} x d x + v x^{2} d v - 2 vx d x - 2 x^{2} d v

= v^{2} x d x + v x^{2} d v - 2 vx d x - 2 x^{2} d v

e^{\frac{1}{2} (1 - x^{2})}

\frac{( x + 7 ) ( x - 4 )}{x - 2}

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

(2 x y + x^{2} - 1) d y

a (e^{- k x} + e^{k x})

What is expand (vx-2x)(vdx+xdv) ?
The solution to expand (vx-2x)(vdx+xdv) is v^2xdx+vx^2dv-2vxdx-2x^2dv