Question

Use implicit differentiation to find the slope of the tangent line to the curve defined by xy6+6xy=42xy6+6xy=42 at the point (6,1)(6,1).

the slope of the tangent line to the curve at the given point is

Answer 1

The given curve is xy⁶ + 6xy = 42.

Perform the implicit differentiation.
y⁶ + 6xy⁵y' + 6y + 6xy' = 0
y'(6xy⁵ + 6x) + y⁶ + 6y = 0

`y'=- (y^(6)+6y)/(6xy^(5)+6x )`

At the point (6,1), the slope is

`y'(6,1)=- (1+6)/(6(6)+6(6))=- (7)/(72)`

Answer:
`- (7)/(72)`