Question

What is the second derivative of xy+y^2=4

Answer 1

Given function xy+y^2 = 4

Apply derivative to each side wth respect of x.

d/dx(xy+y^2) = 4

Remember the formula d/dx(uv) = uv'+vu'

xdy/dx+y+2ydy/dx = 0

dy/dx(x+2y)+y = 0

Subtract y from each side.

dy/dx(x+2y) = -y

Divide to each side by (x+2y).

dy/dx = -y/(x+2y) = y'

Again derivative to each side with respect of x.

d/dx(dy/dx) =  d/dx(-y/(x+2y))

Remember the formula d/dx(u/v) = (vu'-uv')/v^2

d^2y/dx^2 = -[(x+2y)dy/dx+y/dx(x+2y)]/(x+2y)^2

y'' = -[(x+2y)dy/dx+y+2dy/dx]/(x+2y)^2

y'' = -[dy/dx(x+2y+2)+y]/(x+2y)^2