First find #(dy)/(dx)# by implicitly differentiating #x^2+4y^2=5#:

#x^2+4y^2=5#

#2x+8y((dy)/(dx))=0#

#((dy)/(dx))(8y) = -2x#

#(dy)/(dx) = (-2x)/(8y)#

#(dy)/(dx) = (-x)/(4y)#

Now implicitly differentiate #(dy)/(dx)#:

Use the quotient rule:

#(d^2y)/(dx^2) = [(4y)(-1)-(-x)(4(dy/dx))]/((4y)^2)#

Simplify:

#(d^2y)/(dx^2) = [-4y+4x(dy/dx)]/(16y^2)#

Substitute #(dy)/(dx) =(-x)/(4y)#

#(d^2y)/(dx^2)= [-cancel(4)y+cancel(4)x((-x)/(4y))]/(4cancel(16)y^2)#

Simplify:

#(d^2y)/(dx^2)= [(-y)/(4y^2)]-[(x^2)/((4y)(4y^2))]#

#(d^2y)/(dx^2)= [(-1)/(4y)]-[(x^2)/(16y^3)]#

Make a common denominator to combine into one fraction:

#(d^2y)/(dx^2)= (-4y^2-x^2)/(16y^3)#

#(d^2y)/(dx^2)= (-(4y^2+x^2))/(16y^3)#

Recall that the original equation states that #4y^2+x^2=5#, so:

#(d^2y)/(dx^2)= (-5)/(16y^3)#