To determine

To find: The derivative y′′ by implicit differentiation.

Answer

The second derivative of the equation is y′′=−14y3.

Explanation

Given:

The equation x2+4y2=4.

Derivative rules:

(1) Chain rule: If y=f(u) and u=g(x)  are both differentiable function, then dydx=dydu⋅dudx.

(2) Quotient Rule: If f1(x) and f2(x) are both differentiable, then ddx[f1(x)f2(x)]=f2(x)ddx[f1(x)]−f1(x)ddx[f2(x)][f2x]2.

Calculation:

Obtain the first derivative of the equation.

x2+4y2=4

Differentiate the equation implicitly with respect to x,

ddx(x2+4y2)=ddx(4)ddx(x2)+4ddx(y2)=0

Apply the chain rule (1) and simplify the terms,

2x+(4ddy(y2)dydx)=02x+(4(2y)dydx)=02x+8ydydx=0dydx=−x4y

Therefore, the derivative of the equation is dydx=−x4y.

Obtain the second derivative of the equation.

The first derivative is y′=−x4y.

Differentiate the above equation implicitly with respect to x,

y′′=ddx(y′)=ddx(−x4y)

Apply the quotient rule (2) and simplify the terms,

y′′=−4yddx(x)−xddx(4y)(4y)2=−4yddx(x)−4xddx(y)(4y)2=−4y(1)−4xdydx16y2=−y−xdydx4y2

Substitute dydx=−x4y,

y′′=−y−x(−x4y)4y2=−y+x24y4y2=−4y2+x24y(4y2)=−4y2+x216y3

Substitute x2+4y2=4 and simplify the terms,

y′′=−416y3=−14y3

Therefore, the second derivative of the equation is y′′=−14y3.