First find (dy)/(dx)dydx by implicitly differentiating x^2+4y^2=5x2+4y2=5:
x^2+4y^2=5x2+4y2=5
2x+8y((dy)/(dx))=02x+8y(dydx)=0
((dy)/(dx))(8y) = -2x(dydx)(8y)=−2x
(dy)/(dx) = (-2x)/(8y)dydx=−2x8y
(dy)/(dx) = (-x)/(4y)dydx=−x4y
Now implicitly differentiate (dy)/(dx)dydx:
Use the quotient rule:
(d^2y)/(dx^2) = [(4y)(-1)-(-x)(4(dy/dx))]/((4y)^2)d2ydx2=(4y)(−1)−(−x)(4(dydx))(4y)2
Simplify:
(d^2y)/(dx^2) = [-4y+4x(dy/dx)]/(16y^2)d2ydx2=−4y+4x(dydx)16y2
Substitute (dy)/(dx) =(-x)/(4y)dydx=−x4y
(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)
