Could you please help me with this calculus problem?

Question

Using implicit differenctiation find the slope of the curve xy^3+xy=8 at the point (4,1)

Answer 1

The curve equation xy³ + xy = 8

Differentiate implicitly with respect to x.

3xy²y' + y³ + xy'+ y = 0

3xy²y'+ xy' = - y³ - y

y'(3xy²+ x) = - y³ - y

y' = (- y³ - y)/(3xy²+ x)

Substitute the values (x , y ) = (4, 1) in above equation.

y' = (- 1³ - 1)/[3(4)(1)² + 4]

y' = (- 1 - 1)/[12 + 4]

y' = -2/16

y' = -1/8

(-1/8) is the slope of tangent line to the curve at (4, 1).