Find y'' by implicit differentiation.

Question

4x^3 + 5y^3 = 6?

Answer 1

The  function is 4x³ + 5y³ = 6

Differentiate the function with respect to x .

4*3x² + 5*3y² y' = 0                           [ derivative of constant = 0 and derivative of xⁿ = n x^n-1 ]

12 x² + 15y² y' = 0

15y² y' = -12 x² 

y' = -12 x² / 15y²

y' = -4x² / 5y²

Now Differentiate the function y' with respect to x  .

y'' =  [5y² (-4*2x) - (-4x²)(5*2y)] / (5y²)²                     [ derivative of  u/v = (vu' - uv') / v² ]        

 y'' =  [-40y²x + 40x²y] / (5y²)²

 y'' =  40xy[-y + x] / 25y^4

 y'' =  8xy[ x - y] / 5y^4

 y'' =  8x[ x - y] / 5y³

So the double differentiation of 4x³ + 5y³ = 6 is y'' =  8x[ x - y] / 5y³ .