Question

Find `(d^2y)/(dx^2)∣_[(x,y)=(2,1)]` if y is a differentiable function of `x` satisfying the equation `x^3+2y^3 = 5xy`? Plot it?

Answer 1

`125/16`

Explanation:

Taking the derivative of `x^3+2y^3=5xy` with respect to `x`, we see that

`3x^2+6y^2dy/dx=5y+5xdy/dx`

Rearranging and solving for `dy/dx` shows that

`dy/dx=(5y-3x^2)/(6y^2-5x)`

Note that

`(dy)/(dx)∣_[(x,y)=(2,1)]=(5-12)/(6-10)=7/4`

Differentiating once more:

`(d^2y)/dx^2=((5ydy/dx-6x)(6y^2-5x)-(12ydy/dx-5)(5y-3x^2))/(6y^2-5x)^2`

When evaluating this at `(2,1)`, each instance of `dy/dx` becomes `7/4`.

`(d^2y)/dx^2|_[(x,y)=(2,1)]=((5(7/4)-12)(6-10)-(12(7/4)-5)(5-12))/(6-10)^2`

`(d^2y)/dx^2|_[(x,y)=(2,1)]=((-13/4)(-4)-16(-7))/16`

`(d^2y)/dx^2|_[(x,y)=(2,1)]=(13+112)/16=125/16`