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.