Question

How do you use implicit differentiation to find the slope of the curve given `xy^5+x^5y=1` at (-1,-1)?

Answer 1

Since `(-1,-1)` is not a point on the given curve, this question has no solution as it stands. However the slope of `xy^5+x^5y=2` at `(-1,-1)` is -1.

Explanation:

Differentiate both sides of the equation `xy^5+x^5y=2` with respect to `x` leads to

`{dx}/{dx} y^5+x {dy^5}/{dx}+{dx^5}/{dx} y+x^5 {dy}/{dx} = 0`

or

`y^5 +5xy^4 {dy}/{dx} +5x^4y+x^5 {dy}/{dx} = 0`

So that at `(-1,-1)` we have

`(-1)^5+5(-1)(-1)^4 {dy}/{dx}+5(-1)^4(-1)+(-1)^5 {dy}/{dx}=0`

or

`-6 {dy}/{dx} -6=0`

so that the slope is -1.