First, we can find #dy/dx# by differentiating both sides.
#x^2+4y^2=4#
#d/dx(x^2)+d/dx(4y^2)=d/dx(4)#
#2x+(4*2y)*dy/dx=0#
#8y*dy/dx=-2x#
#dy/dx=(-2x)/(8y)#
#dy/dx=(-x)/(4y)#
Then, we can differentiate it again to get the second derivative.
#d/dx (dy/dx)=d/dx((-x)/(4y))#
#=[4y*d/dx(-x)-(-x)*d/dx(4y)]/(4y)^2#
#=[4y*(-1)-(-x)*4dy/dx]/(16y^2)#
#=[-4y+4xdy/dx]/(16y^2)#
Then, we can sub #dy/dx#that we found in the above and get the final answer.
#=[-4y+4x((-x)/(4y))]/(16y^2)#
#=[-4y-x^2/y]/(16y^2)#
#=[-4y^2-x^2]/(16y^3)#
Here is the answer. Hope this can help you :)
