Question

How do you implicitly differentiate `x^4+2y^2=8`?

Answer 1

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

Explanation:

`(dx^4)/(dx) = 4x^3`

`(d2y^2)/(dx) = (d2y^2)/(dy)*(dy)/(dx) = 4y(dy)/(dx)`

`(d (x^4+2y^2))/(dx) = (dx^4)/(dx)+(d2y^2)/(dx) = 4x^3+4y(dy)/(dx)`

`(d8)/(dx) = 0`

Since `x^4+2y^2=8`
`color(white)("XXX")(d(x^4+2y^2))/(dx) = (d8)/(dx)`

`color(white)("XXX")4x^3+4y(dy)/(dx)=0`

`color(white)("XXX")4y(dy)/(dx) = -4x^3`

`color(white)("XXX")(dy)/(dx) = (-4x^3)/(4y)=-x^3/y`