Question

Suppose x and y are related by the given equation and use implicit differentiation to determine dydx.x7y+y7x=7.

Answer 1

Looks like the equation is

`x^7y+y^7x=7`

Differentiate both sides with respect to
`x`, taking
`y` to be a function of
`x`.

`(\mathrm d[x^7y+y^7x])/(\mathrm dx)=(\mathrm d[7])/(\mathrm dx)`

`(\mathrm d[x^7])/(\mathrm dx)y+x^7(\mathrm dy)/(\mathrm dx)+(\mathrm d[y^7])/(\mathrm dx)x+y^7(\mathrm dx)/(\mathrm dx)=0`

`7x^6y+x^7(\mathrm dy)/(\mathrm dx)+7y^6x(\mathrm dy)/(\mathrm dx)+y^7=0`

Solve for
`(\mathrm dy)/(\mathrm dx)`:

`(x^7+7y^6)(\mathrm dy)/(\mathrm dx)=-(7x^6y+y^7)`

`(\mathrm dy)/(\mathrm dx)=-(7x^6y+y^7)/(x^7+7y^6)`