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

Question

Cameron Alexander · Accepted Answer

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.

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.

Elijah Abram · Answer

undefined To find the slope of the curve at the point (-1, -1) using implicit differentiation, follow these steps:

1. Differentiate both sides of the equation with respect to x.
2. Treat y as a function of x and apply the chain rule when differentiating terms involving y.
3. After differentiation, solve for dy/dx, the derivative of y with respect to x.
4. Substitute the point (-1, -1) into the expression for dy/dx to find the slope at that point.

The expression for dy/dx will give you the slope of the curve at the given point (-1, -1).