What is expand (-vx+2x)(v+x(dv)/(dx)) ?

The solution to expand (-vx+2x)(v+x(dv)/(dx)) is -v^2x-vx^2(dv)/(dx)+2vx+2x^2(dv)/(dx)

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expand (-vx+2x)(v+x(dv)/(dx))

expand

(- vx + 2 x) (v + x \frac{d v}{d x})

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

Solution steps

(- vx + 2 x) (v + x \frac{d v}{d x})

Apply FOIL method:

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

a = - vx, b = 2 x, c = v, d = x \frac{d v}{d x}

= (- vx) v + (- vx) x \frac{d v}{d x} + 2 xv + 2 xx \frac{d v}{d x}

Apply minus-plus rules

+ (- a) = - a

= - vvx - vxx \frac{d v}{d x} + 2 vx + 2 xx \frac{d v}{d x}

Simplify

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

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

(x + 2) d x + (y - 2) d y

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

(- x)^{3}

(3 c e^{3 x} - 5 d e^{- 5 x})^{2}

(a + b + c + d + 1)^{3}

What is expand (-vx+2x)(v+x(dv)/(dx)) ?
The solution to expand (-vx+2x)(v+x(dv)/(dx)) is -v^2x-vx^2(dv)/(dx)+2vx+2x^2(dv)/(dx)