Given x^2+ 8y^2=8 find d^2y/dx^2

Question

Find the second derivative of the equation. 

Answer 1

x² + 8y² = 8

 

2x + 16y(dy/dx) = 0

 

So, dy/dx = -2x/(16y) = -x/(8y)

 

d²y/dx² = [-1(8y) - 8(dy/dx)(-x)]/(64y²)

 

            = [-8y +8x(-x/(8y))] / (64y²) = [-8y - x²/y] / (64y²)

 

                                                      = (-8y² - x²) / (64y³) = -8 / (64y³)

 

                                                                                      = -1/(8y³)

 

Answer 2

First derivative:

 

 2x + 16y y' = 0 <--- power rule and chain rule

 

Second derivative:

 2 + 16y y'' + 16y'^2 = 0 <--- product rule

 

1 + 8y y'' + 8 y'^2 = 0 <--- factors out the 2

 

y'' = (-1 - 8 y' ^2 )/(8y)