Question

If x^2+3y^2=8+2xy, then d^2y/dx^2 at (2,2) =

Answer 1

We are given
x² + 3y² = 8 + 2xy
We are asked for
d^2y/dx^2 at (2,2)
Which is the second derivative of the expression in terms of x evaluated at (2,2)

The expression can't be express explicitly in terms of x
Doing implicit differentiation
2x dx + 6y dy = 2y dx + 2 x dy
Solving for dy/dx
dy/dx = (2x - 2y) / (2x - 6y)
Then,
d^2y/dx^2 at (2,2) can now be determined

Answer 2

The value of
`(d^2y)/(dx^2)` at (2,2) is
`(-1)/(4)`.

Explanation:

The given equation is

`x^2+3y^2=8+2xy`

Differentiate with respect to x.

`2x+3(2yy')=0+2(xy'+y)`

Here,
`y'=(dy)/(dx)`

`2(x+3yy')=2(xy'+y)`

`x+3yy'=xy'+y`

`3yy'-xy'=y-x`

`(3y-x)y'=y-x`

`y'=(y-x)/(3y-x)`

The value of first derivative at (2,2) is

`y'_((2,2))=(2-2)/(3(2)-2)=0`

Differentiate with respect to x. Using quotient rule we get,

`y''=((3y-x)(y'-1)-(y-x)(3y'-1))/((3y-x)^2)`

The value of second derivative at (0,0) is

`y''_((2,2))=((3(2)-2)(y'_((2,2))-1)-(2-2)(3y'_((2,2))-1))/((3(2)-(2))^2)`

`y''_((2,2))=((4)(0-1)-0)/(4^2)`

`y''_((2,2))=(-4)/(16)`

`y''_((2,2))=(-1)/(4)`

Therefore the value of
`(d^2y)/(dx^2)` at (2,2) is
`(-1)/(4)`.