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)#
